Numerical Asymptotic Solutions Of Differential Equations
Thurston, Gaylen A.
1992-01-01
Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.
Numerical solution of Boltzmann's equation
International Nuclear Information System (INIS)
Sod, G.A.
1976-04-01
The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig
Automatic validation of numerical solutions
DEFF Research Database (Denmark)
Stauning, Ole
1997-01-01
This thesis is concerned with ``Automatic Validation of Numerical Solutions''. The basic theory of interval analysis and self-validating methods is introduced. The mean value enclosure is applied to discrete mappings for obtaining narrow enclosures of the iterates when applying these mappings...... differential equations, but in this thesis, we describe how to use the methods for enclosing iterates of discrete mappings, and then later use them for discretizing solutions of ordinary differential equations. The theory of automatic differentiation is introduced, and three methods for obtaining derivatives...... are described: The forward, the backward, and the Taylor expansion methods. The three methods have been implemented in the C++ program packages FADBAD/TADIFF. Some examples showing how to use the three metho ds are presented. A feature of FADBAD/TADIFF not present in other automatic differentiation packages...
Spurious Numerical Solutions Of Differential Equations
Lafon, A.; Yee, H. C.
1995-01-01
Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.
Exact solutions, numerical relativity and gravitational radiation
International Nuclear Information System (INIS)
Winicour, J.
1986-01-01
In recent years, there has emerged a new use for exact solutions to Einstein's equation as checks on the accuracy of numerical relativity codes. Much has already been written about codes based upon the space-like Cauchy problem. In the case of two Killing vectors, a numerical characteristic initial value formulation based upon two intersecting families of null hypersurfaces has successfully evolved the Schwarzschild and the colliding plane wave vacuum solutions. Here the author discusses, in the context of exact solutions, numerical studies of gravitational radiation based upon the null cone initial value problem. Every stage of progress in the null cone approach has been associated with exact solutions in some sense. He begins by briefly recapping this history. Then he presents two new examples illustrating how exact solutions can be useful
Numerically satisfactory solutions of Kummer recurrence relations
J. Segura (Javier); N.M. Temme (Nico)
2008-01-01
textabstractPairs of numerically satisfactory solutions as $n\\rightarrow \\infty$ for the three-term recurrence relations satisfied by the families of functions $_1\\mbox{F}_1(a+\\epsilon_1 n; b +\\epsilon_2 n;z)$, $\\epsilon_i \\in {\\mathbb Z}$, are given. It is proved that minimal solutions always
Analysis of numerical solutions for Bateman equations
International Nuclear Information System (INIS)
Loch, Guilherme G.; Bevilacqua, Joyce S.
2013-01-01
The implementation of stable and efficient numerical methods for solving problems involving nuclear transmutation and radioactive decay chains is the main scope of this work. The physical processes associated with irradiations of samples in particle accelerators, or the burning spent nuclear fuel in reactors, or simply the natural decay chains, can be represented by a set of first order ordinary differential equations with constant coefficients, for instance, the decay radioactive constants of each nuclide in the chain. Bateman proposed an analytical solution for a particular case of a linear chain with n nuclides decaying in series and with different decay constants. For more complex and realistic applications, the construction of analytical solutions is not viable and the introduction of numerical techniques is imperative. However, depending on the magnitudes of the decay radioactive constants, the matrix of coefficients could be almost singular, generating unstable and non convergent numerical solutions. In this work, different numerical strategies for solving systems of differential equations were implemented, the Runge-Kutta 4-4, Adams Predictor-Corrector (PC2) and the Rosenbrock algorithm, this last one more specific for stiff equations. Consistency, convergence and stability of the numerical solutions are studied and the performance of the methods is analyzed for the case of the natural decay chain of Uranium-235 comparing numerical with analytical solutions. (author)
Numerical solution of large sparse linear systems
International Nuclear Information System (INIS)
Meurant, Gerard; Golub, Gene.
1982-02-01
This note is based on one of the lectures given at the 1980 CEA-EDF-INRIA Numerical Analysis Summer School whose aim is the study of large sparse linear systems. The main topics are solving least squares problems by orthogonal transformation, fast Poisson solvers and solution of sparse linear system by iterative methods with a special emphasis on preconditioned conjuguate gradient method [fr
Constructing exact symmetric informationally complete measurements from numerical solutions
Appleby, Marcus; Chien, Tuan-Yow; Flammia, Steven; Waldron, Shayne
2018-04-01
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs to their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here, we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of an SIC. Using this method, we have calculated 69 new exact solutions, including nine new dimensions, where previously only numerical solutions were known—which more than triples the number of known exact solutions. In some cases, the solutions require number fields with degrees as high as 12 288. We use these solutions to confirm that they obey the number-theoretic conjectures, and address two questions suggested by the previous work.
New numerical method for solving the solute transport equation
International Nuclear Information System (INIS)
Ross, B.; Koplik, C.M.
1978-01-01
The solute transport equation can be solved numerically by approximating the water flow field by a network of stream tubes and using a Green's function solution within each stream tube. Compared to previous methods, this approach permits greater computational efficiency and easier representation of small discontinuities, and the results are easier to interpret physically. The method has been used to study hypothetical sites for disposal of high-level radioactive waste
International Nuclear Information System (INIS)
Kaya, Dogan; El-Sayed, Salah M.
2003-01-01
In this Letter we present an Adomian's decomposition method (shortly ADM) for obtaining the numerical soliton-like solutions of the potential Kadomtsev-Petviashvili (shortly PKP) equation. We will prove the convergence of the ADM. We obtain the exact and numerical solitary-wave solutions of the PKP equation for certain initial conditions. Then ADM yields the analytic approximate solution with fast convergence rate and high accuracy through previous works. The numerical solutions are compared with the known analytical solutions
On the numerical solution of fault trees
International Nuclear Information System (INIS)
Demichela, M.; Piccinini, N.; Ciarambino, I.; Contini, S.
2003-01-01
In this paper an account will be given of the numerical solution of the logic trees directly extracted from the Recursive Operability Analysis. Particular attention will be devoted to the use of the NOT and INH logic gates for correct logical representation of Fault Trees prior to their quantitative resolution. The NOT gate is needed for correct logical representation of events when both non-intervention and correct intervention of a protective system may lead to a Top Event. The INH gate must be used to correctly represent the time link between two events that are both necessary, but must occur in sequence. Some numerical examples will be employed to show both the correct identification of the events entering the INH gates and how use of the AND gate instead of the INH gate leads to overestimation of the probability of occurrence of a Top Event
Numerical solutions of the Vlasov equation
International Nuclear Information System (INIS)
Satofuka, Nobuyuki; Morinishi, Koji; Nishida, Hidetoshi
1985-01-01
A numerical procedure is derived for the solutions of the one- and two-dimensional Vlasov-Poisson system equations. This numerical procedure consists of the phase space discretization and the integration of the resulting set of ordinary differential equations. In the phase space discretization, derivatives with respect to the phase space variable are approximated by a weighted sum of the values of the distribution function at properly chosen neighboring points. Then, the resulting set of ordinary differential equations is solved by using an appropriate time integration scheme. The results for linear Landau damping, nonlinear Landau damping and counter-streaming plasmas are investigated and compared with those of the splitting scheme. The proposed method is found to be very accurate and efficient. (author)
Numerical double layer solutions with ionization
International Nuclear Information System (INIS)
Andersson, D.; Soerensen, J.
1982-08-01
Maxwell's equation div D = ro in one dimension is solved numerically, taking ionization into account. Time independent anode sheath and double layer solutions are obtained. By varying voltage, neutral gas pressure, temperature of the trapped ions on the cathode side and density and temperature of the trapped electrones on the anode side, diagrams are constructed that show permissible combinations of these parameters. Results from a recent experiment form a subset. Distribution functions, the Langmuir condition, some scaling laws and a possible application to the lower ionosphere are discussed. (Authors)
Numerical solutions of diffusive logistic equation
International Nuclear Information System (INIS)
Afrouzi, G.A.; Khademloo, S.
2007-01-01
In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains. The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the 'finite difference scheme' will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years
Numerical solution of the radionuclide transport equation
International Nuclear Information System (INIS)
Hadermann, J.; Roesel, F.
1983-11-01
A numerical solution of the one-dimensional geospheric radionuclide chain transport equation based on the pseudospectral method is developed. The advantages of this approach are flexibility in incorporating space and time dependent migration parameters, arbitrary boundary conditions and solute rock interactions as well as efficiency and reliability. As an application the authors investigate the impact of non-linear sorption isotherms on migration in crystalline rock. It is shown that non-linear sorption, in the present case a Freundlich isotherm, may reduce concentration at the geosphere outlet by orders of magnitude provided the migration time is comparable or larger than the half-life of the nuclide in question. The importance of fixing dispersivity within the continuum approach is stressed. (Auth.)
Sensitivity analysis of numerical solutions for environmental fluid problems
International Nuclear Information System (INIS)
Tanaka, Nobuatsu; Motoyama, Yasunori
2003-01-01
In this study, we present a new numerical method to quantitatively analyze the error of numerical solutions by using the sensitivity analysis. If a reference case of typical parameters is one calculated with the method, no additional calculation is required to estimate the results of the other numerical parameters such as more detailed solutions. Furthermore, we can estimate the strict solution from the sensitivity analysis results and can quantitatively evaluate the reliability of the numerical solution by calculating the numerical error. (author)
Numerical solution of the polymer system
Energy Technology Data Exchange (ETDEWEB)
Haugse, V.; Karlsen, K.H.; Lie, K.-A.; Natvig, J.R.
1999-05-01
The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Remain solutions and exact tracking of waves. It is well known that the front tracking method may introduce a blow-up of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time that decreases with the discretization parameter. For multidimensional problems, front tracking is combined with dimensional splitting and numerical experiments indicate that large splitting steps can be used without loss of accuracy. Typical CFL numbers are in the range of 10 to 20 and comparisons with the Riemann free, high-resolution method confirm the high efficiency of front tracking. The polymer system, coupled with an elliptic pressure equation, models two-phase, tree-component polymer flooding in an oil reservoir. Two examples are presented where this model is solved by a sequential time stepping procedure. Because of the approximate Riemann solver, the method is non-conservative and CFL members must be chosen only moderately larger than unity to avoid substantial material balance errors generated in near-well regions after water breakthrough. Moreover, it is demonstrated that dimensional splitting may introduce severe grid orientation effects for unstable displacements that are accentuated for decreasing discretization parameters. 9 figs., 2 tabs., 26 refs.
Numerical solution for heave of expansive soils
International Nuclear Information System (INIS)
Sadrnezhad, S. A.
1999-01-01
A numerical solution for heave prediction is developed within the context theories for both saturated and unsaturated soil behaviors. Basically, lowering the potential level of compressing on a saturated layer will cause heaving due to water absorption. This water absorption is in an opposite way, similar to water dissipation as what happens during unloading in consolidation process. However, in unsaturated layers any change of the stability of potential energy level will cause the tendency of change in particle interconnection forces. So, any change by either distressing or the variation of moisture ratio will lead to soil heave. In this paper a finite element solution is employed for predicting the heave in saturated soil similar to unloading in consolidation. Also, in the case of unsaturated soil, equivalent soil suction as negative pore water pressures in applied to soil elements as equivalent nodal forces. To show the potential of this method, test results were com pated with those obtained from computations. These comparisons show that the presented method is capable of predicting the heave phenomenon quite well
Numerical integration of asymptotic solutions of ordinary differential equations
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
The numerical solution of ICRF fields in axisymmetric mirrors
International Nuclear Information System (INIS)
Phillips, M.W.; Todd, A.M.M.
1986-01-01
The numerics of a numerical code called GARFIELD (Grumman Aerospace RF fIELD code) designed to calculate the three-dimensional structure of ICRF fields in axisymmetric mirrors is presented. The code solves the electromagnetic wave equation for the electric field using a cold plasma dispersion relation with a small collision term to simulate absorption. The full wave solution including E.B is computed. The fields are Fourier analyzed in the poloidal direction and solved on a grid in the axial and radial directions. A two-dimensional equilibrium can be used as the source of equilibrium data. This allows us to extend previous studies of ICRF wave propagation and absorption in mirrors to include the effect of axial variation of the magnetic field and density. (orig.)
Numerical simulation of the shot peening process under previous loading conditions
International Nuclear Information System (INIS)
Romero-Ángeles, B; Urriolagoitia-Sosa, G; Torres-San Miguel, C R; Molina-Ballinas, A; Benítez-García, H A; Vargas-Bustos, J A; Urriolagoitia-Calderón, G
2015-01-01
This research presents a numerical simulation of the shot peening process and determines the residual stress field induced into a component with a previous loading history. The importance of this analysis is based on the fact that mechanical elements under shot peening are also subjected to manufacturing processes, which convert raw material into finished product. However, material is not provided in a virgin state, it has a previous loading history caused by the manner it is fabricated. This condition could alter some beneficial aspects of the residual stress induced by shot peening and could accelerate the crack nucleation and propagation progression. Studies were performed in beams subjected to strain hardening in tension (5ε y ) before shot peening was applied. Latter results were then compared in a numerical assessment of an induced residual stress field by shot peening carried out in a component (beam) without any previous loading history. In this paper, it is clearly shown the detrimental or beneficial effect that previous loading history can bring to the mechanical component and how it can be controlled to improve the mechanical behavior of the material
Numerical solution of ordinary differential equations
Fox, L
1987-01-01
Nearly 20 years ago we produced a treatise (of about the same length as this book) entitled Computing methods for scientists and engineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence of the 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety of computational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized books on each major topic: for example on numerical linear algebra, numerical approximation, numeri...
Rotationally symmetric numerical solutions to the sine-Gordon equation
DEFF Research Database (Denmark)
Olsen, O. H.; Samuelsen, Mogens Rugholm
1981-01-01
We examine numerically the properties of solutions to the spherically symmetric sine-Gordon equation given an initial profile which coincides with the one-dimensional breather solution and refer to such solutions as ring waves. Expanding ring waves either exhibit a return effect or expand towards...
Numerical solution of the multichannel scattering problem
International Nuclear Information System (INIS)
Korobov, V.I.
1992-01-01
A numerical algorithm for solving the multichannel elastic and inelastic scattering problem is proposed. The starting point is the system of radial Schroedinger equations with linear boundary conditions imposed at some point R=R m placed somewhere in asymptotic region. It is discussed how the obtained linear equation can be splitted into a zero-order operator and its pertturbative part. It is shown that Lentini - Pereyra variable order finite-difference method appears to be very suitable for solving that kind of problems. The derived procedure is applied to dμ+t→tμ+d inelastic scattering in the framework of the adiabatic multichannel approach. 19 refs.; 1 fig.; 1 tab
Dynamics of the east India coastal current. 2. Numerical solutions
Digital Repository Service at National Institute of Oceanography (India)
McCreary, J.P.; Han, W.; Shankar, D.; Shetye, S.R.
A linear, continuously stratified model is used to investigate the dynamics of the East India Coastal Current (EICC). Solutions are found numerically in a basin that resembles the Indian Ocean basin north of 29 degrees S, and they are forced...
On numerical solution of Burgers' equation by homotopy analysis method
International Nuclear Information System (INIS)
Inc, Mustafa
2008-01-01
In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions
Six-dimensional localized black holes: Numerical solutions
International Nuclear Information System (INIS)
Kudoh, Hideaki
2004-01-01
To test the strong-gravity regime in Randall-Sundrum braneworlds, we consider black holes bound to a brane. In a previous paper, we studied numerical solutions of localized black holes whose horizon radii are smaller than the AdS curvature radius. In this paper, we improve the numerical method and discuss properties of the six-dimensional (6D) localized black holes whose horizon radii are larger than the AdS curvature radius. At a horizon temperature T≅1/2πl, the thermodynamics of the localized black hole undergo a transition with its character changing from a 6D Schwarzschild black hole type to a 6D black string type. The specific heat of the localized black holes is negative, and the entropy is greater than or nearly equal to that of the 6D black strings with the same thermodynamic mass. The large localized black holes show flattened horizon geometries, and the intrinsic curvature of the horizon four-geometry becomes negative near the brane. Our results indicate that the recovery mechanism of lower-dimensional Einstein gravity on the brane works even in the presence of the black holes
On the numerical solution of the neutron fractional diffusion equation
International Nuclear Information System (INIS)
Maleki Moghaddam, Nader; Afarideh, Hossein; Espinosa-Paredes, Gilberto
2014-01-01
Highlights: • The new version of neutron diffusion equation which established on the fractional derivatives is presented. • The Neutron Fractional Diffusion Equation (NFDE) is solved in the finite differences frame. • NFDE is solved using shifted Grünwald-Letnikov definition of fractional operators. • The results show that “K eff ” strongly depends on the order of fractional derivative. - Abstract: In order to core calculation in the nuclear reactors there is a new version of neutron diffusion equation which is established on the fractional partial derivatives, named Neutron Fractional Diffusion Equation (NFDE). In the NFDE model, neutron flux in each zone depends directly on the all previous zones (not only on the nearest neighbors). Under this circumstance, it can be said that the NFDE has the space history. We have developed a one-dimension code, NFDE-1D, which can simulate the reactor core using arbitrary exponent of differential operators. In this work a numerical solution of the NFDE is presented using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame. The model is validated with some numerical experiments where different orders of fractional derivative are considered (e.g. 0.999, 0.98, 0.96, and 0.94). The results show that the effective multiplication factor (K eff ) depends strongly on the order of fractional derivative
Solution of Milne problem by Laplace transformation with numerical inversion
International Nuclear Information System (INIS)
Campos Velho, H.F. de.
1987-12-01
The Milne problem for monoenergetic neutrons, by Laplace Transform of the neutron transport integral equation with numerical inversion of the transformed solution by gaussian quadrature, using the fatorization of the dispersion function. The resulted is solved compared its analitical solution. (author) [pt
Exact and numerical solutions of generalized Drinfeld-Sokolov equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya36@yahoo.com
2008-04-14
In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM)
Exact and numerical solutions of generalized Drinfeld-Sokolov equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2008-01-01
In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM)
Numerical solution of electrostatic problems of the accelerator project VICKSI
International Nuclear Information System (INIS)
Janetzki, U.
1975-03-01
In this work, the numerical solution to a few of the electrostatic problems is dealt with which have occured within the framework of the heavy ion accelerator project VICKSI. By means of these selected examples, the versatile applicability of the numerical method is to be demonstrated, and simultaneously assistance is given for the solution of similar problems. The numerical process for solving ion-optics problems consists generally of two steps. In the first step, the potential distribution for a given boundary value problem is iteratively calculated for the Laplace equation, and then the image characteristics of the electostatic lense are investigated using the Raytrace method. (orig./LH) [de
Numerical solution of non-linear diffusion problems
International Nuclear Information System (INIS)
Carmen, A. del; Ferreri, J.C.
1998-01-01
This paper presents a method for the numerical solution of non-linear diffusion problems using finite-differences in moving grids. Due to the presence of steep fronts in the solution domain and to the presence of advective terms originating in the grid movement, an implicit TVD scheme, first order in time and second order in space has been developed. Some algebraic details of the derivation are given. Results are shown for the pure advection of a scalar as a test case and an example dealing with the slow spreading of viscous fluids over plane surfaces. The agreement between numerical and analytical solutions is excellent. (author). 8 refs., 3 figs
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Introduction to the numerical solutions of Markov chains
Stewart, Williams J
1994-01-01
A cornerstone of applied probability, Markov chains can be used to help model how plants grow, chemicals react, and atoms diffuse - and applications are increasingly being found in such areas as engineering, computer science, economics, and education. To apply the techniques to real problems, however, it is necessary to understand how Markov chains can be solved numerically. In this book, the first to offer a systematic and detailed treatment of the numerical solution of Markov chains, William Stewart provides scientists on many levels with the power to put this theory to use in the actual world, where it has applications in areas as diverse as engineering, economics, and education. His efforts make for essential reading in a rapidly growing field. Here, Stewart explores all aspects of numerically computing solutions of Markov chains, especially when the state is huge. He provides extensive background to both discrete-time and continuous-time Markov chains and examines many different numerical computing metho...
Case studies in the numerical solution of oscillatory integrals
International Nuclear Information System (INIS)
Adam, G.
1992-06-01
A numerical solution of a number of 53,249 test integrals belonging to nine parametric classes was attempted by two computer codes: EAQWOM (Adam and Nobile, IMA Journ. Numer. Anal. (1991) 11, 271-296) and DO1ANF (Mark 13, 1988) from the NAG library software. For the considered test integrals, EAQWOM was found to be superior to DO1ANF as it concerns robustness, reliability, and friendly user information in case of failure. (author). 9 refs, 3 tabs
Numerical solution of dynamic equilibrium models under Poisson uncertainty
DEFF Research Database (Denmark)
Posch, Olaf; Trimborn, Timo
2013-01-01
We propose a simple and powerful numerical algorithm to compute the transition process in continuous-time dynamic equilibrium models with rare events. In this paper we transform the dynamic system of stochastic differential equations into a system of functional differential equations of the retar...... solution to Lucas' endogenous growth model under Poisson uncertainty are used to compute the exact numerical error. We show how (potential) catastrophic events such as rare natural disasters substantially affect the economic decisions of households....
Efficient numerical solution to vacuum decay with many fields
Energy Technology Data Exchange (ETDEWEB)
Masoumi, Ali; Olum, Ken D.; Shlaer, Benjamin, E-mail: ali@cosmos.phy.tufts.edu, E-mail: kdo@cosmos.phy.tufts.edu, E-mail: shlaer@cosmos.phy.tufts.edu [Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155 (United States)
2017-01-01
Finding numerical solutions describing bubble nucleation is notoriously difficult in more than one field space dimension. Traditional shooting methods fail because of the extreme non-linearity of field evolution over a macroscopic distance as a function of initial conditions. Minimization methods tend to become either slow or imprecise for larger numbers of fields due to their dependence on the high dimensionality of discretized function spaces. We present a new method for finding solutions which is both very efficient and able to cope with the non-linearities. Our method directly integrates the equations of motion except at a small number of junction points, so we do not need to introduce a discrete domain for our functions. The method, based on multiple shooting, typically finds solutions involving three fields in around a minute, and can find solutions for eight fields in about an hour. We include a numerical package for Mathematica which implements the method described here.
Numerical solution of distributed order fractional differential equations
Katsikadelis, John T.
2014-02-01
In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.
A numerical guide to the solution of the bidomain equations of cardiac electrophysiology
Pathmanathan, Pras
2010-06-01
Simulation of cardiac electrical activity using the bidomain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bidomain modelling. Each component of bidomain simulations-discretisation, ODE-solution, linear system solution, and parallelisation-is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid to the choice of stimulus currents, compatibility conditions for the equations, the solution of singular linear systems, and convergence of the numerical scheme. © 2010 Elsevier Ltd.
A numerical guide to the solution of the bidomain equations of cardiac electrophysiology
Pathmanathan, Pras; Bernabeu, Miguel O.; Bordas, Rafel; Cooper, Jonathan; Garny, Alan; Pitt-Francis, Joe M.; Whiteley, Jonathan P.; Gavaghan, David J.
2010-01-01
Simulation of cardiac electrical activity using the bidomain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bidomain modelling. Each component of bidomain simulations-discretisation, ODE-solution, linear system solution, and parallelisation-is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid to the choice of stimulus currents, compatibility conditions for the equations, the solution of singular linear systems, and convergence of the numerical scheme. © 2010 Elsevier Ltd.
Numerical solution of the resistive magnetohydrodynamic boundary-layer equations
International Nuclear Information System (INIS)
Glasser, A.H.; Jardin, S.C.; Tesauro, G.
1983-10-01
Three different techniques are presented for numerical solution of the equations governing the boundary layer of resistive magnetohydrodynamic tearing and interchange instabilities in toroidal geometry. Excellent agreement among these methods and with analytical results provides confidence in the correctness of the results. Solutions obtained in regimes where analytical medthods fail indicate a new scaling for the tearing mode as well as the existence of a new regime of stability
Comparing numerical methods for the solutions of the Chen system
International Nuclear Information System (INIS)
Noorani, M.S.M.; Hashim, I.; Ahmad, R.; Bakar, S.A.; Ismail, E.S.; Zakaria, A.M.
2007-01-01
In this paper, the Adomian decomposition method (ADM) is applied to the Chen system which is a three-dimensional system of ODEs with quadratic nonlinearities. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the classical fourth-order Runge-Kutta (RK4) numerical solutions are made. In particular we look at the accuracy of the ADM as the Chen system changes from a non-chaotic system to a chaotic one. To highlight some computational difficulties due to a high Lyapunov exponent, a comparison with the Lorenz system is given
On mesh refinement and accuracy of numerical solutions
Zhou, Hong; Peters, Maria; van Oosterom, Adriaan
1993-01-01
This paper investigates mesh refinement and its relation with the accuracy of the boundary element method (BEM) and the finite element method (FEM). TO this end an isotropic homogeneous spherical volume conductor, for which the analytical solution is available, wag used. The numerical results
LED-based Photometric Stereo: Modeling, Calibration and Numerical Solutions
DEFF Research Database (Denmark)
Quéau, Yvain; Durix, Bastien; Wu, Tao
2018-01-01
We conduct a thorough study of photometric stereo under nearby point light source illumination, from modeling to numerical solution, through calibration. In the classical formulation of photometric stereo, the luminous fluxes are assumed to be directional, which is very difficult to achieve in pr...
The Numerical Solution of an Abelian Ordinary Differential Equation ...
African Journals Online (AJOL)
In this paper we present a relatively new technique call theNew Hybrid of Adomian decomposition method (ADM) for solution of an Abelian Differential equation. The numerical results of the equation have been obtained in terms of convergent series with easily computable component. These methods are applied to solve ...
Numerical Solution of Differential Algebraic Equations and Applications
DEFF Research Database (Denmark)
Thomsen, Per Grove
2005-01-01
These lecture notes have been written as part of a special course on the numerical solution of Differential Algebraic Equations and applications . The course was held at IMM in the spring of 2005. The authors of the different chapters have all taken part in the course and the chapters are written...
Numerical solution of field theories using random walks
International Nuclear Information System (INIS)
Barnes, T.; Daniell, G.J.
1985-01-01
We show how random walks in function space can be employed to evaluate field theoretic vacuum expectation values numerically. Specific applications which we study are the two-point function, mass gap, magnetization and classical solutions. This technique offers the promise of faster calculations using less computer memory than current methods. (orig.)
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.; Al-Juhani, Amnah
2015-01-01
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
A numerical solution of the coupled proton-H atom transport equations for the proton aurora
International Nuclear Information System (INIS)
Basu, B.; Jasperse, J.R.; Grossbard, N.J.
1990-01-01
A numerical code has been developed to solve the coupled proton-H atom linear transport equations for the proton aurora. The transport equations have been simplified by using plane-parallel geometry and the forward-scattering approximations only. Otherwise, the equations and their numerical solutions are exact. Results are presented for the particle fluxes and the energy deposition rates, and they are compared with the previous analytical results that were obtained by using additional simplifying approximations. It is found that although the analytical solutions for the particle fluxes differ somewhat from the numerical solutions, the energy deposition rates calculated by the two methods agree to within a few percent. The accurate particle fluxes given by the numerical code are useful for accurate calculation of the characteristic quantities of the proton aurora, such as the ionization rates and the emission rates
Spurious solutions in few-body equations. II. Numerical investigations
International Nuclear Information System (INIS)
Adhikari, S.K.
1979-01-01
A recent analytic study of spurious solutions in few-body equations by Adhikari and Gloeckle is here complemented by numerical investigations. As proposed by Adhikari and Gloeckle we study numerically the spurious solutions in the three-body Weinberg type equations and draw some general conclusions about the existence of spurious solutions in three-body equations with the Weinberg kernel and in other few-body formulations. In particular we conclude that for most of the potentials we encounter in problems of nuclear physics the three-body Weinberg type equation will not have a spurious solution which may interfere with the bound state or scattering calculation. Hence, if proven convenient, the three-body Weinberg type equation can be used in practical calculations. The same conclusion is true for the three-body channel coupling array scheme of Kouri, Levin, and Tobocman. In the case of the set of six coupled four-body equations proposed by Rosenberg et al. and the set of the Bencze-Redish-Sloan equations a careful study of the possible spurious solutions is needed before using these equations in practical calculations
Performance analysis of numeric solutions applied to biokinetics of radionuclides
International Nuclear Information System (INIS)
Mingatos, Danielle dos Santos; Bevilacqua, Joyce da Silva
2013-01-01
Biokinetics models for radionuclides applied to dosimetry problems are constantly reviewed by ICRP. The radionuclide trajectory could be represented by compartmental models, assuming constant transfer rates between compartments. A better understanding of physiological or biochemical phenomena, improve the comprehension of radionuclide behavior in the human body and, in general, more complex compartmental models are proposed, increasing the difficulty of obtaining the analytical solution for the system of first order differential equations. Even with constant transfer rates numerical solutions must be carefully implemented because of almost singular characteristic of the matrix of coefficients. In this work we compare numerical methods with different strategies for ICRP-78 models for Thorium-228 and Uranium-234. The impact of uncertainty in the parameters of the equations is also estimated for local and global truncation errors. (author)
International Nuclear Information System (INIS)
Wehner, M.F.
1983-01-01
A path-integral solution is derived for processes described by nonlinear Fokker-Plank equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is show t give accurate results. 35 refs., 5 figs
Numerical solution of a reaction-diffusion equation
International Nuclear Information System (INIS)
Moyano, Edgardo A.; Scarpettini, Alberto F.
2000-01-01
The purpose of the present work to continue the observations and the numerical experiences on a reaction-diffusion model, that is a simplified form of the neutronic flux equation. The model is parabolic, nonlinear, with Dirichlet boundary conditions. The purpose is to approximate non trivial solutions, asymptotically stables for t → ∞, that is solutions that tend to the elliptic problem, in the Lyapunov sense. It belongs to the so-called reaction-diffusion equations of semi linear kind, that is, linear equations in the heat operator and they have a nonlinear reaction function, in this case f (u, a, b) = u (a - b u), being u concentration, a and b parameters. The study of the incidence of these parameters take an interest to the neutronic flux physics. So that we search non trivial, positive and bounded solutions. The used algorithm is based on the concept of monotone and ordered sequences, and on the existence theorem of Amann and Sattinger. (author)
Numerical solution for identification of feedback coefficients in nuclear reactors
International Nuclear Information System (INIS)
Ebizuka, Yoshie; Sakai, Hideo
1975-01-01
Quasilinearization technique was studied to determine the Kinetic parameters of nuclear reactors. The method of solution was generalized to the determination of the parameters contained in a nonlinear system with nonlinear boundary conditions. A computer program, SNR-3, was developed to solve the resulting nonlinear two-point boundary value equations with generalized boundary conditions. In this paper, the problem formulation and the method of solution are explained for a general type of time dependent problem. A flow chart shows the procedure of numerical solution. The method was then applied to the determination of the critical factor and the reactivity feedback coefficients of reactors to investigate the accuracy and the applicability of the present method. The results showed that the present method was considerably successful, but that the random observation error effected the results of the identification. (Aoki, K.)
Numerical benchmarking of SPEEDUP trademark against point kinetics solutions
International Nuclear Information System (INIS)
Gregory, M.V.
1993-02-01
SPEEDUP trademark is a state-of-the-art, dynamic, chemical process modeling package offered by Aspen Technology. In anticipation of new customers' needs for new analytical tools to support the site's waste management activities, SRTC has secured a multiple-user license to SPEEDUP trademark. In order to verify both the installation and mathematical correctness of the algorithms in SPEEDUP trademark, we have performed several numerical benchmarking calculations. These calculations are the first steps in establishing an on-site quality assurance pedigree for SPEEDUP trademark. The benchmark calculations consisted of SPEEDUP trademark Version 5.3L representations of five neutron kinetics benchmarks (each a mathematically stiff system of seven coupled ordinary differential equations), whose exact solutions are documented in the open literature. In all cases, SPEEDUP trademark solutions to be in excellent agreement with the reference solutions. A minor peculiarity in dealing with a non-existent discontinuity in the OPERATION section of the model made itself evident
International Nuclear Information System (INIS)
Esmail, S.F.H.
2011-01-01
The mathematical formulation of numerous physical problems a results in differential equations actually partial or ordinary differential equations.In our study we are interested in solutions of partial differential equations.The aim of this work is to calculate the concentrations of the pollution, by solving the atmospheric diffusion equation(ADE) using different mathematical methods of solution. It is difficult to solve the general form of ADE analytically, so we use some assumptions to get its solution.The solutions of it depend on the eddy diffusivity profiles(k) and the wind speed u. We use some physical assumptions to simplify its formula and solve it. In the present work, we solve the ADE analytically in three dimensions using Green's function method, Laplace transform method, normal mode method and these separation of variables method. Also, we use ADM as a numerical method. Finally, comparisons are made with the results predicted by the previous methods and the observed data.
Numerical solution of a model for a superconductor field problem
International Nuclear Information System (INIS)
Alsop, L.E.; Goodman, A.S.; Gustavson, F.G.; Miranker, W.L.
1979-01-01
A model of a magnetic field problem occurring in connection with Josephson junction devices is derived, and numerical solutions are obtained. The model is of mathematical interest, because the magnetic vector potential satisfies inhomogeneous Helmholtz equations in part of the region, i.e., the superconductors, and the Laplace equation elsewhere. Moreover, the inhomogeneities are the guage constants for the potential, which are different for each superconductor, and their magnitudes are proportional to the currents flowing in the superconductors. These constants are directly related to the self and mutual inductances of the superconducting elements in the device. The numerical solution is obtained by the iterative use of a fast Poisson solver. Chebyshev acceleration is used to reduce the number of iterations required to obtain a solution. A typical problem involves solving 100,000 simultaneous equations, which the algorithm used with this model does in 20 iterations, requiring three minutes of CPU time on an IBM VM/370/168. Excellent agreement is obtained between calculated and observed values for the inductances
Random ordinary differential equations and their numerical solution
Han, Xiaoying
2017-01-01
This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs). RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems. They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor ...
Numerical solution of plasma fluid equations using locally refined grids
International Nuclear Information System (INIS)
Colella, P.
1997-01-01
This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasma under consideration is typical of those used for the processing of semiconductors. The governing equations consist of a drift-diffusion model of the electrons and an isothermal model of the ions coupled by Poisson's equation. A discretization of the equations is given for a uniform spatial grid, and a time-split integration scheme is developed. The algorithm is then extended to accommodate locally refined grids. This extension involves the advancement of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. A brief discussion of a software implementation is followed by a presentation of numerical results
CSR Fields: Direct Numerical Solution of the Maxwell's Equation
International Nuclear Information System (INIS)
Novokhatski, Alexander
2011-01-01
We discuss the properties of the coherent electromagnetic fields of a very short, ultra-relativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersion-free time-domain algorithm which employs a more efficient use of finite element mesh techniques and hence produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the CSR fields including coherent edge radiation. This approach should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources. The coherent synchrotron radiation (CSR) fields have a strong action on the beam dynamics of very short bunches, which are moving in the bends of all kinds of magnetic elements. They are responsible for additional energy loss and energy spread; micro bunching and beam emittance growth. These fields may bound the efficiency of damping rings, electron-positron colliders and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned. This is relevant to most high-brightness beam applications. On the other hand these fields together with transition radiation fields can be used for beam diagnostics or even as a powerful resource of THz radiation. A history of the study of CSR and a good collection of references can be found in (1). Electromagnetic theory suggests several methods on how to calculate CSR fields. The most popular method is to use Lienard-Wiechert potentials. Other approach is to solve numerically the approximate equations, which are a Schrodinger type equation. These numerical methods are described in (2). We suggest that a direct solution of Maxwell's equations together with Newton's equations can describe the detailed structure of the CSR fields (3).
Numerical solution of modified differential equations based on symmetry preservation.
Ozbenli, Ersin; Vedula, Prakash
2017-12-01
In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.
Numerical solution of the Schroedinger equation with a polynomial potential
International Nuclear Information System (INIS)
Campoy, G.; Palma, A.
1986-01-01
A numerical method for solving the Schroedinger equation for a potential expressed as a polynomial is proposed. The basic assumption relies on the asymptotic properties of the solution of this equation. It is possible to obtain the energies and the stationary state functions simultaneously. They analyze, in particular, the cases of the quartic anharmonic oscillator and a hydrogen atom perturbed by a quadratic term, obtaining its energy eigenvalues for some values of the perturbation parameter. Together with the Hellmann-Feynman theorem, they use their algorithm to calculate expectation values of x'' for arbitrary positive values of n. 4 tables
A note on numerical solution to the problem of criticality
International Nuclear Information System (INIS)
Kyncl, J.
2002-01-01
The contribution deals with numerical solution to the problem of criticality for neutron transport equation by the external source iteration method. Especially, the speed of convergence is examined. It is shown that if neutron absorption in the medium considered is high and if the space region occupied by the medium is large then a slow convergence of the iterations can be expected. This expectation is confirmed by results to CB4 benchmark obtained by MCNP code. Besides the results presented some questions concerning applications of them to criticality calculations are pointed out (Author)
Numerical solutions of differential equations of an ionization chamber
International Nuclear Information System (INIS)
Novkovic, D.; Tomasevic, M.; Subotic, K.; Manic, S.
1998-01-01
A system of reduced differential equations generally valid for plane-parallel, cylindrical, and spherical ionization chambers filled with air, which is appropriate for numerical solution, has been derived. The system has been solved for all three geometries. The comparison of the calculated results of Armstrong and Tate, for plane-parallel ionization chambers, and Sprinkle and Tate, for spherical ionization chambers, with the present calculations has shown a good agreement. The calculated values for ionization chambers filled with CO 2 were also in good agreement with the experimental data of Moriuchi et al (author)
Numerical solution of High-kappa model of superconductivity
Energy Technology Data Exchange (ETDEWEB)
Karamikhova, R. [Univ. of Texas, Arlington, TX (United States)
1996-12-31
We present formulation and finite element approximations of High-kappa model of superconductivity which is valid in the high {kappa}, high magnetic field setting and accounts for applied magnetic field and current. Major part of this work deals with steady-state and dynamic computational experiments which illustrate our theoretical results numerically. In our experiments we use Galerkin discretization in space along with Backward-Euler and Crank-Nicolson schemes in time. We show that for moderate values of {kappa}, steady states of the model system, computed using the High-kappa model, are virtually identical with results computed using the full Ginzburg-Landau (G-L) equations. We illustrate numerically optimal rates of convergence in space and time for the L{sup 2} and H{sup 1} norms of the error in the High-kappa solution. Finally, our numerical approximations demonstrate some well-known experimentally observed properties of high-temperature superconductors, such as appearance of vortices, effects of increasing the applied magnetic field and the sample size, and the effect of applied constant current.
A numerical solution of a singular boundary value problem arising in boundary layer theory.
Hu, Jiancheng
2016-01-01
In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner-Skan equation. And the one-dimensional third-order boundary value problem on interval [Formula: see text] is equivalently transformed into a second-order boundary value problem on finite interval [Formula: see text]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.
International Nuclear Information System (INIS)
Khotylev, V.A.; Hoogenboom, J.E.
1996-01-01
The paper presents new techniques for the solution of the nuclear reactor equation in diffusion approximation, that has enhanced efficiency and stability. The code system based on the new technique solves a number of steady-state and/or transient problems with coupled thermal hydraulics in one-, two-, or three dimensional geometry with reduced CPU time as compared to similar code systems of previous generations if well-posed neutronics problems are considered. Automated detection of ill-posed problem and selection of the appropriate numerical method makes the new code system capable of yielding a correct solution for wider range of problems without user intervention. (author)
Energy Technology Data Exchange (ETDEWEB)
Khotylev, V.A.; Hoogenboom, J.E. [Delft Univ. of Technology, Interfaculty Reactor Inst., Delft (Netherlands)
1996-07-01
The paper presents new techniques for the solution of the nuclear reactor equation in diffusion approximation, that has enhanced efficiency and stability. The code system based on the new technique solves a number of steady-state and/or transient problems with coupled thermal hydraulics in one-, two-, or three dimensional geometry with reduced CPU time as compared to similar code systems of previous generations if well-posed neutronics problems are considered. Automated detection of ill-posed problem and selection of the appropriate numerical method makes the new code system capable of yielding a correct solution for wider range of problems without user intervention. (author)
Multiresolution strategies for the numerical solution of optimal control problems
Jain, Sachin
There exist many numerical techniques for solving optimal control problems but less work has been done in the field of making these algorithms run faster and more robustly. The main motivation of this work is to solve optimal control problems accurately in a fast and efficient way. Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. The algorithm adapted dynamically to any existing or emerging irregularities in the solution by automatically allocating more grid points to the region where the solution exhibited sharp features and fewer points to the region where the solution was smooth. Thereby, the computational time and memory usage has been reduced significantly, while maintaining an accuracy equivalent to the one obtained using a fine uniform mesh. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a
Numerical and analytical solutions for problems relevant for quantum computers
International Nuclear Information System (INIS)
Spoerl, Andreas
2008-01-01
Quantum computers are one of the next technological steps in modern computer science. Some of the relevant questions that arise when it comes to the implementation of quantum operations (as building blocks in a quantum algorithm) or the simulation of quantum systems are studied. Numerical results are gathered for variety of systems, e.g. NMR systems, Josephson junctions and others. To study quantum operations (e.g. the quantum fourier transform, swap operations or multiply-controlled NOT operations) on systems containing many qubits, a parallel C++ code was developed and optimised. In addition to performing high quality operations, a closer look was given to the minimal times required to implement certain quantum operations. These times represent an interesting quantity for the experimenter as well as for the mathematician. The former tries to fight dissipative effects with fast implementations, while the latter draws conclusions in the form of analytical solutions. Dissipative effects can even be included in the optimisation. The resulting solutions are relaxation and time optimised. For systems containing 3 linearly coupled spin (1)/(2) qubits, analytical solutions are known for several problems, e.g. indirect Ising couplings and trilinear operations. A further study was made to investigate whether there exists a sufficient set of criteria to identify systems with dynamics which are invertible under local operations. Finally, a full quantum algorithm to distinguish between two knots was implemented on a spin(1)/(2) system. All operations for this experiment were calculated analytically. The experimental results coincide with the theoretical expectations. (orig.)
Numerical solution of a flow inside a labyrinth seal
Directory of Open Access Journals (Sweden)
Šimák Jan
2012-04-01
Full Text Available The aim of this study is a behaviour of a ﬂow inside a labyrinth seal on a rotating shaft. The labyrinth seal is a type of a non-contact seal where a leakage of a ﬂuid is prevented by a rather complicated path, which the ﬂuid has to overcome. In the presented case the sealed medium is the air and the seal is made by a system of 20 teeth on a rotating shaft situated against a smooth static surface. Centrifugal forces present due to the rotation of the shaft create vortices in each chamber and thus dissipate the axial velocity of the escaping air.The structure of the ﬂow ﬁeld inside the seal is studied through the use of numerical methods. Three-dimensional solution of the Navier-Stokes equations for turbulent ﬂow is very time consuming. In order to reduce the computational time we can simplify our problem and solve it as an axisymmetric problem in a two-dimensional meridian plane. For this case we use a transformation of the Navier-Stokes equations and of the standard k-omega turbulence model into a cylindrical coordinate system. A ﬁnite volume method is used for the solution of the resulting problem. A one-side modiﬁcation of the Riemann problem for boundary conditions is used at the inlet and at the outlet of the axisymmetric channel. The total pressure and total density (temperature are to be used preferably at the inlet whereas the static pressure is used at the outlet for the compatibility. This idea yields physically relevant boundary conditions. The important characteristics such as a mass ﬂow rate and a power loss, depending on a pressure ratio (1.1 - 4 and an angular velocity (1000 - 15000 rpm are evaluated.
Milne, a routine for the numerical solution of Milne's problem
Rawat, Ajay; Mohankumar, N.
2010-11-01
The routine Milne provides accurate numerical values for the classical Milne's problem of neutron transport for the planar one speed and isotropic scattering case. The solution is based on the Case eigen-function formalism. The relevant X functions are evaluated accurately by the Double Exponential quadrature. The calculated quantities are the extrapolation distance and the scalar and the angular fluxes. Also, the H function needed in astrophysical calculations is evaluated as a byproduct. Program summaryProgram title: Milne Catalogue identifier: AEGS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 701 No. of bytes in distributed program, including test data, etc.: 6845 Distribution format: tar.gz Programming language: Fortran 77 Computer: PC under Linux or Windows Operating system: Ubuntu 8.04 (Kernel version 2.6.24-16-generic), Windows-XP Classification: 4.11, 21.1, 21.2 Nature of problem: The X functions are integral expressions. The convergence of these regular and Cauchy Principal Value integrals are impaired by the singularities of the integrand in the complex plane. The DE quadrature scheme tackles these singularities in a robust manner compared to the standard Gauss quadrature. Running time: The test included in the distribution takes a few seconds to run.
Born approximation to a perturbative numerical method for the solution of the Schroedinger equation
International Nuclear Information System (INIS)
Adam, Gh.
1978-01-01
A step function perturbative numerical method (SF-PN method) is developed for the solution of the Cauchy problem for the second order liniar differential equation in normal form. An important point stressed in the present paper, which seems to have been previously ignored in the literature devoted to the PN methods, is the close connection between the first order perturbation theory of the PN approach and the wellknown Born approximation, and, in general, the connection between the varjous orders of the PN corrections and the Neumann series. (author)
The numerical solution of boundary value problems over an infinite domain
International Nuclear Information System (INIS)
Shepherd, M.; Skinner, R.
1976-01-01
A method is presented for the numerical solution of boundary value problems over infinite domains. An example that illustrates also the strength and accuracy of a numerical procedure for calculating Green's functions is described in detail
Discrete convolution-operators and radioactive disintegration. [Numerical solution
Energy Technology Data Exchange (ETDEWEB)
Kalla, S L; VALENTINUZZI, M E [UNIVERSIDAD NACIONAL DE TUCUMAN (ARGENTINA). FACULTAD DE CIENCIAS EXACTAS Y TECNOLOGIA
1975-08-01
The basic concepts of discrete convolution and discrete convolution-operators are briefly described. Then, using the discrete convolution - operators, the differential equations associated with the process of radioactive disintegration are numerically solved. The importance of the method is emphasized to solve numerically, differential and integral equations.
Numeric databases on the kinetics of transient species in solution
International Nuclear Information System (INIS)
Helman, W.P.; Hug, G.L.; Carmichael, Ian; Ross, A.B.
1988-01-01
A description is given of data compilations on the kinetics of transient species in solution. In particular information is available for the reactions of radicals in aqueous solution and for excited states such as singlet molecular oxygen and those of metal complexes in solution. Methods for compilation and use of the information in computer-readable form are also described. Emphasis is placed on making the database available for online searching. (author)
Numerical solution of the kinetic equation in reactor shielding
International Nuclear Information System (INIS)
Germogenova, T.A.
1975-01-01
A review is made of methods of solving marginal problems of multi-group systems of equations of neutron and γ radiation transfer. The first stage of the solution - the quantification of the basic task, is determined by the qualitative behaviour of the solution - is the nature of its performance and asymptotics. In the second stage - solution of the approximating system, various modifications of the iterative method are as a rule used. A description is given of the features of the major Soviet complexes of programmes (ROZ and RADUGA) for the solution of multi-group systems of transfer equations and some methodological research findings are presented. (author)
Numerical solutions of the N-body problem
International Nuclear Information System (INIS)
Marciniak, A.
1985-01-01
Devoted to the study of numerical methods for solving the general N-body problem and related problems, this volume starts with an overview of the conventional numerical methods for solving the initial value problem. The major part of the book contains original work and features a presentation of special numerical methods conserving the constants of motion in the general N-body problem and methods conserving the Jacobi constant in the problem of motion of N bodies in a rotating frame, as well as an analysis of the applications of both (conventional and special) kinds of methods for solving these problems. For all the methods considered, the author presents algorithms which are easily programmable in any computer language. Moreover, the author compares various methods and presents adequate numerical results. The appendix contains PL/I procedures for all the special methods conserving the constants of motion. 91 refs.; 35 figs.; 41 tabs
Numerical Solution of Magnetostatic Field of Maglev System
Directory of Open Access Journals (Sweden)
Jaroslav Sobotka
2008-01-01
Full Text Available The paper deals with the design of the levitation and guidance system of the levitation train Transrapid 08 by means of QuickField 5.0 – a 2D program formagnetic electromagnetic fields solutions.
Numerical solution of fuzzy boundary value problems using Galerkin ...
Indian Academy of Sciences (India)
1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China. 2 Department of ... exact solution of fuzzy first-order boundary value problems. (BVPs). ...... edge partial financial support by the Ministerio de Economıa.
Numerical solution of boundary-integral equations for molecular electrostatics.
Bardhan, Jaydeep P
2009-03-07
Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.
Numerical solution of pipe flow problems for generalized Newtonian fluids
International Nuclear Information System (INIS)
Samuelsson, K.
1993-01-01
In this work we study the stationary laminar flow of incompressible generalized Newtonian fluids in a pipe with constant arbitrary cross-section. The resulting nonlinear boundary value problems can be written in a variational formulation and solved using finite elements and the augmented Lagrangian method. The solution of the boundary value problem is obtained by finding a saddle point of the augmented Lagrangian. In the algorithm the nonlinear part of the equations is treated locally and the solution is obtained by iteration between this nonlinear problem and a global linear problem. For the solution of the linear problem we use the SSOR preconditioned conjugate gradient method. The approximating problem is solved on a sequence of adaptively refined grids. A scheme for adjusting the value of the crucial penalization parameter of the augmented Lagrangian is proposed. Applications to pipe flow and a problem from the theory of capacities are given. (author) (34 refs.)
A numerical solution for a toroidal plasma in equilibrium
International Nuclear Information System (INIS)
Hintz, E.; Sudano, J.P.
1982-01-01
The iterative techniques alternating direction implicit (ADI), sucessive ove-relaxation (SOR) and Gauss-Seidel are applied to a nonlinear elliptical second order differential equation (Grand-Shafranov). This equation was solve with the free boundary conditions plasma-vacuum interface over a rectangular section in cylindrical coordinates R and Z. The current density profile, plasma pressure profile, magnetic and isobaric surfaces are numerically determined for a toroidal plasma in equilibrium. (L.C.) [pt
Numerical Solution of Hamilton-Jacobi Equations in High Dimension
2012-11-23
high dimension FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA-Universita di Roma P. Aldo Moro, 2 00185 ROMA AH930...solution of Hamilton-Jacobi equations in high dimension AFOSR contract n. FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA
Analysis of the Numerical Solution of the Shallow Water Equations
National Research Council Canada - National Science Library
Hamrick, Thomas
1997-01-01
.... The two schemes are finite difference method (FDM) and the finite element method (FEM). After presenting the shallow water equations in several formulations, some examples will be presented. The use of the Fourier transform to find the solution of a semidiscrete analog of the shallow water equations is also demonstrated.
Lectures on the Numerical Solution of Partial Differential Equations.
1981-12-01
Mathematics Rockefeller University Professor F. Brezzi New York, New York 10021 Laboratorio di Analisi Numerica Universita di Pavia Professor Amiram Harten...equations and to control the spacing of the points sj. In the MFE process the grid points move with the solution and cluster atound areas of roughness...149-159. [28] Fichera, G.: Analisi essistenziale per le soluzioni die problemi al contorno misti relativi alle equazione ed ai sistemi di equazioni
New Numerical Solution of von Karman Equation of Lengthwise Rolling
Directory of Open Access Journals (Sweden)
Rudolf Pernis
2015-01-01
Full Text Available The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karman equation was solved by many authors. The solutions reported by authors are used simplifications for solution of Karman equation. The simplifications are based on two cases for approximation of the circular arch: (a by polygonal curve and (b by parabola. The contribution of the present paper for solution of two-dimensional differential equation of rolling is based on description of the circular arch by equation of a circle. The new term relative stress as nondimensional variable was defined. The result from derived mathematical models can be calculated following variables: normal contact stress distribution, front and back tensions, angle of neutral point, coefficient of the arm of rolling force, rolling force, and rolling torque during rolling process. Laboratory cold rolled experiment of CuZn30 brass material was performed. Work hardening during brass processing was calculated. Comparison of theoretical values of normal contact stress with values of normal contact stress obtained from cold rolling experiment was performed. The calculations were not concluded with roll flattening.
Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem
Ceccobello, C.; Farinelli, R.; Titarchuk, L.
2014-01-01
We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the
Numerical solutions of ICRF fields in axisymmetric mirrors
International Nuclear Information System (INIS)
Phillips, M.W.
1985-01-01
The results of a new numerical code called GARFIELD (Grumman Aerospace Rf Field code) that calculates ICRF Fields in axisymmetric mirror geometry (such as the central cell of a tandem mirror or an RF test stand) are presented. The code solves the electromagnetic wave equation using a cold plasma dispersion relation with a small collision frequency to simulate absorption. The purpose of the calculation is to examine how ICRF wave structure and propagation is effected by the axial variation of the magnetic field in a mirror for various antenna designs. In the code the wave equation is solved in flux coordinates using a finite element method. This should allow more complex dielectric tensors to be modeled in the future. The resulting matrix is solved iteratively, to maximize the allowable size of the spatial grid. Results for a typical antenna array in a simple mirror will be shown
Numerical solution of the helmholtz equation for the superellipsoid via the galerkin method
Directory of Open Access Journals (Sweden)
Hy Dinh
2013-01-01
Full Text Available The objective of this work was to find the numerical solution of the Dirichlet problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. In this case we could only obtain good numerical results for super ellipsoids that were more shaped like super cones, which is a narrow range of super ellipsoids. The formula for these shapes was: $x=cos(xsin(y^{n},y=sin(xsin(y^{n},z=cos(y$ where $n$ varied from 0.5 to 4. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Dirichlet boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We are researching the feasibility of obtaining good convergence results for the superellipsoid surface. It was our view that smaller and lighter wave numbers would reduce computational costs associated with obtaining Galerkin coefficients. In addition, we hoped to significantly reduce the number of terms in the infinite series needed to modify the original integral equation, all of which were achieved in the analysis of the superellipsoid in a finite range. We used the Green's theorem to solve the integral equation for the boundary of the surface. Previously, multiple surfaces were used to test this method, such as the sphere, ellipsoid, and perturbation of the sphere, pseudosphere and the oval of Cassini Lin and Warnapala , Warnapala and Morgan .
A numerical dressing method for the nonlinear superposition of solutions of the KdV equation
International Nuclear Information System (INIS)
Trogdon, Thomas; Deconinck, Bernard
2014-01-01
In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg–de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly decaying initial data. The second method is used to compute finite-genus solutions of the KdV equation. The combination of these numerical methods allows for the computation of exact solutions that are asymptotically (quasi-)periodic finite-gap solutions and are a nonlinear superposition of dispersive, soliton and (quasi-)periodic solutions in the finite (x, t)-plane. Such solutions are referred to as superposition solutions. We compute these solutions accurately for all values of x and t. (paper)
Numerical solution of three-dimensional magnetic differential equations
International Nuclear Information System (INIS)
Reiman, A.H.; Greenside, H.S.
1987-02-01
A computer code is described that solves differential equations of the form B . del f = h for a single-valued solution f, given a toroidal three-dimensional divergence-free field B and a single-valued function h. The code uses a new algorithm that Fourier decomposes a given function in a set of flux coordinates in which the field lines are straight. The algorithm automatically adjusts the required integration lengths to compensate for proximity to low order rational surfaces. Applying this algorithm to the Cartesian coordinates defines a transformation to magnetic coordinates, in which the magnetic differential equation can be accurately solved. Our method is illustrated by calculating the Pfirsch-Schlueter currents for a stellarator
Numerical solution of compressible flow equations inside an ejector
International Nuclear Information System (INIS)
Omid khah, M. R.; Navid Famili, M. H.; Jalili Keshtiban, E.
2002-01-01
Ejector is important equipment in the chemical industry. It is mainly used for vaccuming and mixing of flows. In the present work a computer modeling of the flow inside an ejector is used to give a better understanding of the principle of the operation of an ejector. Since the fluid inside an ejector passes through subsonic, sonic and supersonic regimens, the pressure field is used as the controlling variable and the density is found through the constitutive equations. The control volume method with a co-location grid, attached to the boundary is used to discretize the domain. The overall solution is obtained by the SIMPLEC method and to dissociate the pressure and the velocity grid Rhie-Chow interpolation method is employed. A central difference approximation method is used to approximate the density on the elements borders and the upwind approximation is used to correct the density correction factors. Both upwind, quick and minimum gradient methods were used to approximate the momentum variables on the control volumes. The resultant matrices are solved with the tri-diagonal method. The accuracy of the model is checked by simulating a flow regiment in a converging-diverging nozzle, and comparing the results with the available experimental data. The results show that for an inviscid the first order approximation produce as an accurate results as the higher order approximations while it has a better stability. However, for the viscous fluid the second order approximation produces a better understanding of the physics of the problem. The solution also showes that the flow field inside an ejector is a complex one and the shock wave has a great influence on the pressure field especially close to the walls. The upper convective quick method did not converge well in the shock calculations while the slowest descent method had a very stable behavior in the analysis of the shock behavior
Explicit appropriate basis function method for numerical solution of stiff systems
International Nuclear Information System (INIS)
Chen, Wenzhen; Xiao, Hongguang; Li, Haofeng; Chen, Ling
2015-01-01
Highlights: • An explicit numerical method called the appropriate basis function method is presented. • The method differs from the power series method for obtaining approximate numerical solutions. • Two cases show the method is fit for linear and nonlinear stiff systems. • The method is very simple and effective for most of differential equation systems. - Abstract: In this paper, an explicit numerical method, called the appropriate basis function method, is presented. The explicit appropriate basis function method differs from the power series method because it employs an appropriate basis function such as the exponential function, or periodic function, other than a polynomial, to obtain approximate numerical solutions. The method is successful and effective for the numerical solution of the first order ordinary differential equations. Two examples are presented to show the ability of the method for dealing with linear and nonlinear systems of differential equations
Fast numerical solution of KKR-CPA equations: Testing new algorithms
Energy Technology Data Exchange (ETDEWEB)
Bruno, E.; Florio, G.M.; Ginatempo, B.; Giuliano, E.S. (Universita di Messina (Italy))
1994-04-01
Some numerical methods for the solution of KKR-CPA equations are discussed and tested. New, efficient, computational algorithms are proposed, allowing a remarkable reduction of computing time and a good reliability in evaluating spectral quantities. 16 refs., 7 figs.
Gómez-Aguilar, J. F.
2018-03-01
In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams-Bashforth-Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking.
special algorithm for the numerical solution of system of initial value ...
African Journals Online (AJOL)
Nwokem et al.
Science World Journal Vol 12(No 4) 2017 ... Over the years, several researchers have considered the collocation method as a way of generating numerical solutions to ... study problems in mathematics, engineering, computer science and.
Numerical Solution of Stokes Flow in a Circular Cavity Using Mesh-free Local RBF-DQ
DEFF Research Database (Denmark)
Kutanaai, S Soleimani; Roshan, Naeem; Vosoughi, A
2012-01-01
This work reports the results of a numerical investigation of Stokes flow problem in a circular cavity as an irregular geometry using mesh-free local radial basis function-based differential quadrature (RBF-DQ) method. This method is the combination of differential quadrature approximation of der...... in solution of partial differential equations (PDEs).......This work reports the results of a numerical investigation of Stokes flow problem in a circular cavity as an irregular geometry using mesh-free local radial basis function-based differential quadrature (RBF-DQ) method. This method is the combination of differential quadrature approximation...... is applied on a two-dimensional geometry. The obtained results from the numerical simulations are compared with those gained by previous works. Outcomes prove that the current technique is in very good agreement with previous investigations and this fact that RBF-DQ method is an accurate and flexible method...
Karkar , Sami; Vergez , Christophe; Cochelin , Bruno
2012-01-01
International audience; We propose a new approach based on numerical continuation and bifurcation analysis for the study of physical models of instruments that produce self- sustained oscillation. Numerical continuation consists in following how a given solution of a set of equations is modified when one (or several) parameter of these equations are allowed to vary. Several physical models (clarinet, saxophone, and violin) are formulated as nonlinear dynamical systems, whose periodic solution...
Numerical Solution of Inviscid Compressible Steady Flows around the RAE 2822 Airfoil
Kryštůfek, P.; Kozel, K.
2015-05-01
The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Euler equations in 2D compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil. The results are compared with the solution using the software Ansys Fluent 15.0.7.
Numerical solution of ordinary differential equations. For classical, relativistic and nano systems
International Nuclear Information System (INIS)
Greenspan, D.
2006-01-01
An up-to-date survey on numerical solutions with theory, intuition and applications. Ordinary differential equations (ODE) play a significant role in mathematics, physics and engineering sciences, and thus are part of relevant college and university courses. Many problems, however, both traditional and modern, do not possess exact solutions, and must be treated numerically. Usually this is done with software packages, but for this to be efficient requires a sound understanding of the mathematics involved. This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic and nano systems. The examples are carefully explained and compiled into an algorithm, each of which is presented generically, independent of a specific programming language, while each chapter is rounded off with exercises. The text meets the demands of MA200 courses and of the newly created Numerical Solution of Differential Equations courses, making it ideal for both students and lecturers in physics, mathematics, mechanical engineering, electrical engineering, as well as for physicists, mathematicians, engineers, and electrical engineers. From the Contents - Euler's Method - Runge-Kutta Methods - The Method of Taylor Expansions - Large Second Order Systems with Application to Nano Systems - Completely Conservative, Covariant Numerical Methodology - Instability - Numerical Solution of Tridiagonal Linear Algebraic Systems and Related Nonlinear Systems - Approximate Solution of Boundary Value Problems - Special Relativistic Motion - Special Topics - Appendix: Basic Matrix Operations - Bibliography. (orig.) (orig.)
Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method
Directory of Open Access Journals (Sweden)
De-Gang Wang
2012-01-01
Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.
Numerical Solution of Compressible Steady Flows around the RAE 2822 Airfoil
Kryštůfek, P.; Kozel, K.
2014-03-01
The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Navier-Stokes equations in 2D laminar compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil.
Numerical Solution of Compressible Steady Flows around the RAE 2822 Airfoil
Directory of Open Access Journals (Sweden)
Kryštůfek P.
2014-03-01
Full Text Available The article presents results of a numerical solution of subsonic, transonic and supersonic flows described by the system of Navier-Stokes equations in 2D laminar compressible flows around the RAE 2822 airfoil. Authors used FVM multistage Runge-Kutta method to numerically solve the flows around the RAE 2822 airfoil.
Numerical Solution of Compressible Steady Flows around the NACA 0012 Airfoil
Directory of Open Access Journals (Sweden)
Kozel K
2013-04-01
Full Text Available The article presents results of a numerical solution of subsonic and transonic flows described by the system of Navier-Stokes equations in 2D laminar compressible flows around the NACA 0012 airfoil. Authors used Runge-Kutta method to numerically solve the flows around the NACA 0012 airfoil.
Numerical solutions of ordinary and partial differential equations in the frequency domain
International Nuclear Information System (INIS)
Hazi, G.; Por, G.
1997-01-01
Numerical problems during the noise simulation in a nuclear power plant are discussed. The solutions of ordinary and partial differential equations are studied in the frequency domain. Numerical methods by the transfer function method are applied. It is shown that the correctness of the numerical methods is limited for ordinary differential equations in the frequency domain. To overcome the difficulties, step-size selection is suggested. (author)
A numerical solution for a class of time fractional diffusion equations with delay
Directory of Open Access Journals (Sweden)
Pimenov Vladimir G.
2017-09-01
Full Text Available This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ2−α+ h4 in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.
Numerical solution of neutron transport equations in discrete ordinates and slab geometry
International Nuclear Information System (INIS)
Serrano Pedraza, F.
1985-01-01
An unified formalism to solve numerically, between other equation, the neutron transport in discrete ordinates, slab geometry, several energy groups and independents of time, has been developed recently. Such a formalism cover some of the conventional schemes as diamond difference, (WDD) characteristic step (SC) lineal characteristic (LC), quadratic characteristic (QC) and lineal discontinuous. Unified formation gives before hand the convergence order of the previously selected scheme. In fact it allows besides to generate a big amount of numerical schemes, with which is also possible to solve numerical equations as soon as neutron transport. The essential purpose of this work was to solve the neutron transport equations in slab geometry and discrete ordinates considering several energy groups without to take under advisement time dependence based in the above mentioned unified formalism. To reach this purpose it was necesary to design a computer code with the name TNOD1 (Neutron transport in discrete ordinates and 1 dimension) which includes each one of the schemes already pointed out. there exist two numerical schemes, also recently developed, quadratic continuous (QC) and cubic continuous (CN), although covered by unified formalism, it has been possible to include them inside this computer code without make substantial changes in its structure. In chapter I, derivative of neutron transport equation independent of time is taken, for angular flux, including boundary conditions and discontinuity. In chapter II the neutron transport equations are obtained in multigroups, independents of time, for approximation of discrete ordinates. Description of theory related with unified formalism and its relationship with mentioned discretization schemes is presented in chapter III. Chapter IV describes the computer code developed and finally, in chapter V different numerical results obtained with TNOD1 program are shown. In Appendix A theorems and mathematical arguments used
DEFF Research Database (Denmark)
Nouhravesh, Nina; Ahlberg, Gustav; Ghouse, Jonas
2016-01-01
BACKGROUND: Hundreds of genetic variants have been described as disease causing in dilated cardiomyopathy (DCM). Some of these associations are now being questioned. We aimed to identify the prevalence of previously DCM associated variants in the Exome Aggregation Consortium (ExAC), in order...... to identify potentially false-positive DCM variants. METHODS: Variants listed as DCM disease-causing variants in the Human Gene Mutation Database were extracted from ExAC. Pathogenicity predictions for these variants were mined from dbNSFP v 2.9 database. RESULTS: Of the 473 DCM variants listed in HGMD, 148...... (31%) were found in ExAC. The expected number of individuals with DCM in ExAC is 25 based on the prevalence in the general population. Yet, 35 variants were found in more than 25 individuals. In 13 genes, we identified all variants previously associated with DCM; four genes contained variants above...
Numerical solutions of a three-point boundary value problem with an ...
African Journals Online (AJOL)
Numerical solutions of a three-point boundary value problem with an integral condition for a third-order partial differential equation by using Laplace transform method Solutions numeriques d'un probleme pour une classe d'equations differentielles d'ordr.
Numerical study of traveling-wave solutions for the Camassa-Holm equation
International Nuclear Information System (INIS)
Kalisch, Henrik; Lenells, Jonatan
2005-01-01
We explore numerically different aspects of periodic traveling-wave solutions of the Camassa-Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied
Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.
2010-07-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.
Malakpoor, K.; Kaasschieter, E.F.; Huyghe, J.M.R.J.
2007-01-01
The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory [J.M. Huyghe and J.D. Janssen, Int. J. Engng. Sci. 35 (1997) 793-802; K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, Mathematical modelling and numerical solution of swelling of cartilaginous tissues.
Energy Technology Data Exchange (ETDEWEB)
Loch, Guilherme G.; Bevilacqua, Joyce S., E-mail: guiloch@ime.usp.br, E-mail: joyce@ime.usp.br [Universidade de Sao Paulo (IME/USP), Sao Paulo, SP (Brazil). Departamento de Matematica Aplicada. Instituto de Matematica e Estatistica; Hiromoto, Goro; Rodrigues Junior, Orlando, E-mail: rodrijr@ipen.br, E-mail: hiromoto@ipen.br [Instituto de Pesquisas Energeticas e Nucleares (IPEN-CNEN/SP), Sao Paulo, SP (Brazil)
2013-07-01
The implementation of stable and efficient numerical methods for solving problems involving nuclear transmutation and radioactive decay chains is the main scope of this work. The physical processes associated with irradiations of samples in particle accelerators, or the burning spent nuclear fuel in reactors, or simply the natural decay chains, can be represented by a set of first order ordinary differential equations with constant coefficients, for instance, the decay radioactive constants of each nuclide in the chain. Bateman proposed an analytical solution for a particular case of a linear chain with n nuclides decaying in series and with different decay constants. For more complex and realistic applications, the construction of analytical solutions is not viable and the introduction of numerical techniques is imperative. However, depending on the magnitudes of the decay radioactive constants, the matrix of coefficients could be almost singular, generating unstable and non convergent numerical solutions. In this work, different numerical strategies for solving systems of differential equations were implemented, the Runge-Kutta 4-4, Adams Predictor-Corrector (PC2) and the Rosenbrock algorithm, this last one more specific for stiff equations. Consistency, convergence and stability of the numerical solutions are studied and the performance of the methods is analyzed for the case of the natural decay chain of Uranium-235 comparing numerical with analytical solutions. (author)
A global numerical solution of the radial Schroedinger equation by second-order perturbation theory
International Nuclear Information System (INIS)
Adam, G.
1979-01-01
A global numerical method, which uses second-order perturbation theory, is described for the solution of the radial Schroedinger equation. The perturbative numerical (PN) solution is derived in two stages: first, the original potential is approximated by a piecewise continuous parabolic function, and second, the resulting Schroedinger equation is solved on each integration step by second-order perturbation theory, starting with a step function reference approximation for the parabolic potential. We get a manageable PN algorithm, which shows an order of accuracy equal to six in the solution of the original Schroedinger equation, and is very stable against round off errors. (author)
Polyanin, A. D.; Sorokin, V. G.
2017-12-01
The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.
Numerical Solution of the Electron Transport Equation in the Upper Atmosphere
Energy Technology Data Exchange (ETDEWEB)
Woods, Mark Christopher [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Holmes, Mark [Rensselaer Polytechnic Inst., Troy, NY (United States); Sailor, William C [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2017-07-01
A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.
Matching of analytical and numerical solutions for neutron stars of arbitrary rotation
International Nuclear Information System (INIS)
Pappas, George
2009-01-01
We demonstrate the results of an attempt to match the two-soliton analytical solution with the numerically produced solutions of the Einstein field equations, that describe the spacetime exterior of rotating neutron stars, for arbitrary rotation. The matching procedure is performed by equating the first four multipole moments of the analytical solution to the multipole moments of the numerical one. We then argue that in order to check the effectiveness of the matching of the analytical with the numerical solution we should compare the metric components, the radius of the innermost stable circular orbit (R ISCO ), the rotation frequency and the epicyclic frequencies Ω ρ , Ω z . Finally we present some results of the comparison.
Matching of analytical and numerical solutions for neutron stars of arbitrary rotation
Energy Technology Data Exchange (ETDEWEB)
Pappas, George, E-mail: gpappas@phys.uoa.g [Section of Astrophysics, Astronomy, and Mechanics, Department of Physics, University of Athens, Panepistimiopolis Zografos GR15783, Athens (Greece)
2009-10-01
We demonstrate the results of an attempt to match the two-soliton analytical solution with the numerically produced solutions of the Einstein field equations, that describe the spacetime exterior of rotating neutron stars, for arbitrary rotation. The matching procedure is performed by equating the first four multipole moments of the analytical solution to the multipole moments of the numerical one. We then argue that in order to check the effectiveness of the matching of the analytical with the numerical solution we should compare the metric components, the radius of the innermost stable circular orbit (R{sub ISCO}), the rotation frequency and the epicyclic frequencies {Omega}{sub {rho}}, {Omega}{sub z}. Finally we present some results of the comparison.
Numerical solution of singularity-perturbed two-point boundary-value problems
International Nuclear Information System (INIS)
Masenge, R.W.P.
1993-07-01
Physical processes which involve transportation of slowly diffusing substances in a fast-flowing medium are mathematically modelled by so-called singularly-perturbed second order convection diffusion differential equations in which the convective first order terms dominate over the diffusive second order terms. In general, analytical solutions of such equations are characterized by having sharp solution fronts in some sections of the interior and/or the boundary of the domain of solution. The presence of these (usually very narrow) layer regions in the solution domain makes the task of globally approximating such solutions by standard numerical techniques very difficult. In this expository paper we use a simple one-dimensional prototype problem as a vehicle for analysing the nature of the numerical approximation difficulties involved. In the sequel we present, without detailed derivation, two practical numerical schemes which succeed in varying degrees in numerically resolving the layer of the solution to the prototype problem. (author). 3 refs, 1 fig., 1 tab
Numerical solution of second-order stochastic differential equations with Gaussian random parameters
Directory of Open Access Journals (Sweden)
Rahman Farnoosh
2014-07-01
Full Text Available In this paper, we present the numerical solution of ordinary differential equations (or SDEs, from each orderespecially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysisfor second-order equations in specially case of scalar linear second-order equations (damped harmonicoscillators with additive or multiplicative noises. Making stochastic differential equations system from thisequation, it could be approximated or solved numerically by different numerical methods. In the case oflinear stochastic differential equations system by Computing fundamental matrix of this system, it could becalculated based on the exact solution of this system. Finally, this stochastic equation is solved by numericallymethod like E.M. and Milstein. Also its Asymptotic stability and statistical concepts like expectationand variance of solutions are discussed.
Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation
International Nuclear Information System (INIS)
Wei, T; Qin, H H; Shi, R
2008-01-01
In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green's formulation, the problem can be transformed into a moment problem. Then we propose a numerical algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimate and convergence analysis have also been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method
Numerical treatment of elliptic BVP with several solutions and of MHD equilibrium problems
International Nuclear Information System (INIS)
Meyer-Spasche, R.
1975-12-01
It is found out empirically that Newton iteration and difference methods are very suitable for the numerical treatment of elliptic boundary value problems (Lu)(x) = f(x,u(x)) in D c R 2 , u/deltaD = g having several solutions. Some convergence theorems for these methods are presented. Some notable numerical examples are given, including bifurcation diagrams, which are interesting in themselves and show also the applicability of the methods developed. (orig./WB) [de
Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline
Directory of Open Access Journals (Sweden)
Ravi Kanth A.S.V.
2016-01-01
Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.
Numerical solution of fluid-structure interaction represented by human vocal folds in airflow
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Valášek J.
2016-01-01
Full Text Available The paper deals with the human vocal folds vibration excited by the fluid flow. The vocal fold is modelled as an elastic body assuming small displacements and therefore linear elasticity theory is used. The viscous incompressible fluid flow is considered. For purpose of numerical solution the arbitrary Lagrangian-Euler method (ALE is used. The whole problem is solved by the finite element method (FEM based solver. Results of numerical experiments with different boundary conditions are presented.
Numerical solution of fluid-structure interaction represented by human vocal folds in airflow
Valášek, J.; Sváček, P.; Horáček, J.
2016-03-01
The paper deals with the human vocal folds vibration excited by the fluid flow. The vocal fold is modelled as an elastic body assuming small displacements and therefore linear elasticity theory is used. The viscous incompressible fluid flow is considered. For purpose of numerical solution the arbitrary Lagrangian-Euler method (ALE) is used. The whole problem is solved by the finite element method (FEM) based solver. Results of numerical experiments with different boundary conditions are presented.
Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
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Bolandtalat A.
2016-01-01
Full Text Available In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.
Numerical solution of inviscid and viscous laminar and turbulent flow around the airfoil
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Slouka Martin
2016-01-01
Full Text Available This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox k-omega model. Calculations are done for NACA 0012 and RAE 2822 airfoil profile for the different angles of upstream flow. Numerical results are compared and discussed with experimental data.
GHOLAMI, SAEID; BABOLIAN, ESMAIL; JAVIDI, MOHAMMAD
2016-01-01
This paper presents a new numerical approach to solve single and multiterm time fractional diffusion equations. In this work, the space dimension is discretized to the Gauss$-$Lobatto points. We use the normalized Grunwald approximation for the time dimension and a pseudospectral successive integration matrix for the space dimension. This approach shows that with fewer numbers of points, we can approximate the solution with more accuracy. Some examples with numerical results in tables and fig...
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Petráš Ivo
2011-01-01
Full Text Available This paper deals with the fractional-order linear and nonlinear models used in bioengineering applications and an effective method for their numerical solution. The proposed method is based on the power series expansion of a generating function. Numerical solution is in the form of the difference equation, which can be simply applied in the Matlab/Simulink to simulate the dynamics of system. Several illustrative examples are presented, which can be widely used in bioengineering as well as in the other disciplines, where the fractional calculus is often used.
The simulation of solute transport: An approach free of numerical dispersion
International Nuclear Information System (INIS)
Carrera, J.; Melloni, G.
1987-01-01
The applicability of most algorithms for simulation of solute transport is limited either by instability or by numerical dispersion, as seen by a review of existing methods. A new approach is proposed that is free of these two problems. The method is based on the mixed Eulerian-Lagrangian formulation of the mass-transport problem, thus ensuring stability. Advection is simulated by a variation of reverse-particle tracking that avoids the accumulation of interpolation errors, thus preventing numerical dispersion. The algorithm has been implemented in a one-dimensional code. Excellent results are obtained, in comparison with an analytical solution. 36 refs., 14 figs., 1 tab
2nd International Workshop on the Numerical Solution of Markov Chains
1995-01-01
Computations with Markov Chains presents the edited and reviewed proceedings of the Second International Workshop on the Numerical Solution of Markov Chains, held January 16--18, 1995, in Raleigh, North Carolina. New developments of particular interest include recent work on stability and conditioning, Krylov subspace-based methods for transient solutions, quadratic convergent procedures for matrix geometric problems, further analysis of the GTH algorithm, the arrival of stochastic automata networks at the forefront of modelling stratagems, and more. An authoritative overview of the field for applied probabilists, numerical analysts and systems modelers, including computer scientists and engineers.
Solutions manual to accompany An introduction to numerical methods and analysis
Epperson, James F
2014-01-01
A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Second Edition An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how to both construct and evaluate approximations for accuracy and performance, which are key skills in a variety of fields. A wide range of higher-level methods and solutions, including new topics such as the roots of polynomials, sp
Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations
Park, Chan-Hee; Aral, Mustafa M.
2007-06-01
In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovič and de Schepper [Frolkovič, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference
On the numerical evaluation of algebro-geometric solutions to integrable equations
International Nuclear Information System (INIS)
Kalla, C; Klein, C
2012-01-01
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated with real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey–Stewartson and the multi-component nonlinear Schrödinger equations
Criteria for the reliability of numerical approximations to the solution of fluid flow problems
International Nuclear Information System (INIS)
Foias, C.
1986-01-01
The numerical approximation of the solutions of fluid flows models is a difficult problem in many cases of energy research. In all numerical methods implementable on digital computers, a basic question is if the number N of elements (Galerkin modes, finite-difference cells, finite-elements, etc.) is sufficient to describe the long time behavior of the exact solutions. It was shown using several approaches that some of the estimates based on physical intuition of N are rigorously valid under very general conditions and follow directly from the mathematical theory of the Navier-Stokes equations. Among the mathematical approaches to these estimates, the most promising (which can be and was already applied to many other dissipative partial differential systems) consists in giving upper estimates to the fractal dimension of the attractor associated to one (or all) solution(s) of the respective partial differential equations. 56 refs
Use of Green's functions in the numerical solution of two-point boundary value problems
Gallaher, L. J.; Perlin, I. E.
1974-01-01
This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.
TLC scheme for numerical solution of the transport equation on equilateral triangular meshes
International Nuclear Information System (INIS)
Walters, W.F.
1983-01-01
A new triangular linear characteristic TLC scheme for numerically solving the transport equation on equilateral triangular meshes has been developed. This scheme uses the analytic solution of the transport equation in the triangle as its basis. The data on edges of the triangle are assumed linear as is the source representation. A characteristic approach or nodal approach is used to obtain the analytic solution. Test problems indicate that the new TLC is superior to the widely used DITRI scheme for accuracy
A note on numerical solution of a parabolic-Schrödinger equation
Ozdemir, Yildirim; Alp, Mustafa
2016-08-01
In the present study, a nonlocal boundary value problem for a parabolic-Schrödinger equation is considered. The stability estimates for the solution of the given problem is established. The first and second order of difference schemes are presented for approximately solving a specific nonlocal boundary problem. The theoretical statements for the solution of these difference schemes are supported by the result of numerical examples.
The Numerical Solution of the Equilibrium Problem for a Stretchable Elastic Beam
Mehdiyeva, G. Y.; Aliyev, A. Y.
2017-08-01
The boundary value problem under consideration describes the equilibrium of an elastic beam that is stretched or contracted by specified forces. The left end of the beam is free of load, and the right end is rigidly lapped. To solve the problem numerically, an appropriate difference problem is constructed. Solving the difference problem, we obtain an approximate solution of the problem. We estimate the approximate solution of the stated problem.
Numerical Modeling for the Solute Uptake from Groundwater by Plants-Plant Uptake Package
El-Sayed, Amr A.
2006-01-01
A numerical model is presented to describe solute transport in groundwater coupled to sorption by plant roots, translocation into plant stems, and finally evapotranspiration. The conceptual model takes into account both Root Concentration Factor, RCF, and Transpiration Stream Concentration Factor, TSCF for chemicals which are a function of Kow. A similar technique used to simulate the solute transport in groundwater to simulate sorption and plant uptake is used. The mathematical equation is s...
Directory of Open Access Journals (Sweden)
Roman Cherniha
2016-06-01
Full Text Available The nonlinear mathematical model for solute and fluid transport induced by the osmotic pressure of glucose and albumin with the dependence of several parameters on the hydrostatic pressure is described. In particular, the fractional space available for macromolecules (albumin was used as a typical example and fractional fluid void volume were assumed to be different functions of hydrostatic pressure. In order to find non-uniform steady-state solutions analytically, some mathematical restrictions on the model parameters were applied. Exact formulae (involving hypergeometric functions for the density of fluid flux from blood to tissue and the fluid flux across tissues were constructed. In order to justify the applicability of the analytical results obtained, a wide range of numerical simulations were performed. It was found that the analytical formulae can describe with good approximation the fluid and solute transport (especially the rate of ultrafiltration for a wide range of values of the model parameters.
Numerical solution of the ekpyrotic scenario in the moduli space approximation
International Nuclear Information System (INIS)
Soerensen, Torquil MacDonald
2005-01-01
A numerical solution to the equations of motion for the ekpyrotic bulk brane scenario in the moduli space approximation is presented. The visible universe brane has positive tension, and we use a potential that goes to zero exponentially at large distance, and also goes to zero at small distance. In the case considered, no bulk brane, visible brane collision occurs in the solution. This property and the general behavior of the solution is qualitatively the same when the visible brane tension is negative, and for many different parameter choices
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Elmira Ashpazzadeh
2018-04-01
Full Text Available A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diusion equation to the solution of algebraic equations. Because of sparsity of these matrices, this method is computationally very attractive and reduces the CPU time and computer memory. Illustrative examples are included to demonstrate the validity and applicability of the new technique.
A numerical method for finding sign-changing solutions of superlinear Dirichlet problems
Energy Technology Data Exchange (ETDEWEB)
Neuberger, J.M.
1996-12-31
In a recent result it was shown via a variational argument that a class of superlinear elliptic boundary value problems has at least three nontrivial solutions, a pair of one sign and one which sign changes exactly once. These three and all other nontrivial solutions are saddle points of an action functional, and are characterized as local minima of that functional restricted to a codimension one submanifold of the Hilbert space H-0-1-2, or an appropriate higher codimension subset of that manifold. In this paper, we present a numerical Sobolev steepest descent algorithm for finding these three solutions.
Numerical solution of modified fokker-planck equation with poissonian input
Czech Academy of Sciences Publication Activity Database
Náprstek, Jiří; Král, Radomil
2010-01-01
Roč. 17, 3/4 (2010), s. 251-268 ISSN 1802-1484 R&D Projects: GA AV ČR(CZ) IAA200710805; GA ČR(CZ) GA103/09/0094 Institutional research plan: CEZ:AV0Z20710524 Keywords : Fokker-Planck equation * poisson ian exciation * numerical solution * transition effects Subject RIV: JN - Civil Engineering
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Decio Levi
2013-10-01
Full Text Available We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the “first differential approximation” is invariant under the same Lie group as the original ordinary differential equation.
Numerical solutions of stochastic Lotka-Volterra equations via operational matrices
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F. Hosseini Shekarabi
2016-03-01
Full Text Available In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.
Several numerical and analytical solutions of the radiative transfer equation (RTE) for plane albedo were compared for solar light reflection by sea water. The study incorporated the simplest case, that being a semi-infinite one-dimensional plane-parallel absorbing and scattering...
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)
2002-01-01
We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.
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SURE KÖME
2014-12-01
Full Text Available In this paper, we investigated the effect of Magnus Series Expansion Method on homogeneous stiff ordinary differential equations with different stiffness ratios. A Magnus type integrator is used to obtain numerical solutions of two different examples of stiff problems and exact and approximate results are tabulated. Furthermore, absolute error graphics are demonstrated in detail.
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Zhanhua Yu
2011-01-01
convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.
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Zhanhua Yu
2011-01-01
Full Text Available We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.
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Marina Popolizio
2018-01-01
Full Text Available Multiterm fractional differential equations (MTFDEs nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.
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Andrea Lani
2006-01-01
Full Text Available Object-oriented platforms developed for the numerical solution of PDEs must combine flexibility and reusability, in order to ease the integration of new functionalities and algorithms. While designing similar frameworks, a built-in support for high performance should be provided and enforced transparently, especially in parallel simulations. The paper presents solutions developed to effectively tackle these and other more specific problems (data handling and storage, implementation of physical models and numerical methods that have arisen in the development of COOLFluiD, an environment for PDE solvers. Particular attention is devoted to describe a data storage facility, highly suitable for both serial and parallel computing, and to discuss the application of two design patterns, Perspective and Method-Command-Strategy, that support extensibility and run-time flexibility in the implementation of physical models and generic numerical algorithms respectively.
Numerical solution of matrix exponential in burn-up equation using mini-max polynomial approximation
International Nuclear Information System (INIS)
Kawamoto, Yosuke; Chiba, Go; Tsuji, Masashi; Narabayashi, Tadashi
2015-01-01
Highlights: • We propose a new numerical solution of matrix exponential in burn-up depletion calculations. • The depletion calculation with extremely short half-lived nuclides can be done numerically stable with this method. • The computational time is shorter than the other conventional methods. - Abstract: Nuclear fuel burn-up depletion calculations are essential to compute the nuclear fuel composition transition. In the burn-up calculations, the matrix exponential method has been widely used. In the present paper, we propose a new numerical solution of the matrix exponential, a Mini-Max Polynomial Approximation (MMPA) method. This method is numerically stable for burn-up matrices with extremely short half-lived nuclides as the Chebyshev Rational Approximation Method (CRAM), and it has several advantages over CRAM. We also propose a multi-step calculation, a computational time reduction scheme of the MMPA method, which can perform simultaneously burn-up calculations with several time periods. The applicability of these methods has been theoretically and numerically proved for general burn-up matrices. The numerical verification has been performed, and it has been shown that these methods have high precision equivalent to CRAM
International Nuclear Information System (INIS)
Colombant, Denis; Manheimer, Wallace
2010-01-01
Flux limitation and preheat are important processes in electron transport occurring in laser produced plasmas. The proper calculation of both of these has been a subject receiving much attention over the entire lifetime of the laser fusion project. Where nonlocal transport (instead of simple single flux limit) has been modeled, it has always been with what we denote the equivalent diffusion solution, namely treating the transport as only a diffusion process. We introduce here a new approach called the nonlocal source solution and show it is numerically viable for laser produced plasmas. It turns out that the equivalent diffusion solution generally underestimates preheat. Furthermore, the advance of the temperature front, and especially the preheat, can be held up by artificial 'thermal barriers'. The nonlocal source method of solution, on the other hand more accurately describes preheat and can stably calculate the solution for the temperature even if the heat flux is up the gradient.
Identifying generalized Fitzhugh-Nagumo equation from a numerical solution of Hodgkin-Huxley model
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Nikola V. Georgiev
2003-01-01
Full Text Available An analytic time series in the form of numerical solution (in an appropriate finite time interval of the Hodgkin-Huxley current clamped (HHCC system of four differential equations, well known in the neurophysiology as an exact empirical model of excitation of a giant axon of Loligo, is presented. Then we search for a second-order differential equation of generalized Fitzhugh-Nagumo (GFN type, having as a solution the given single component (action potential of the numerical solution. The given time series is used as a basis for reconstructing orders, powers, and coefficients of the polynomial right-hand sides of GFN equation approximately governing the process of action potential. For this purpose, a new geometrical method for determining phase space dimension of the unknown dynamical system (GFN equation and a specific modification of least squares method for identifying unknown coefficients are developed and applied.
Numerical solution of the full potential equation using a chimera grid approach
Holst, Terry L.
1995-01-01
A numerical scheme utilizing a chimera zonal grid approach for solving the full potential equation in two spatial dimensions is described. Within each grid zone a fully-implicit approximate factorization scheme is used to advance the solution one interaction. This is followed by the explicit advance of all common zonal grid boundaries using a bilinear interpolation of the velocity potential. The presentation is highlighted with numerical results simulating the flow about a two-dimensional, nonlifting, circular cylinder. For this problem, the flow domain is divided into two parts: an inner portion covered by a polar grid and an outer portion covered by a Cartesian grid. Both incompressible and compressible (transonic) flow solutions are included. Comparisons made with an analytic solution as well as single grid results indicate that the chimera zonal grid approach is a viable technique for solving the full potential equation.
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Henrique Silva Furtado
2009-09-01
Full Text Available Numerical simulation of solute trapping during solidification, using two phase-field model for dilute binary alloys developed by Kim et al. [Phys. Rev. E, 60, 7186 (1999] and Ramirez et al. [Phys. Rev. E, 69, 05167 (2004] is presented here. The simulations on dilute Cu-Ni alloy are in good agreement with one dimensional analytic solution of sharp interface model. Simulation conducted under small solidification velocity using solid-liquid interface thickness (2λ of 8 nanometers reproduced the solute (Cu equilibrium partition coefficient. The spurious numerical solute trapping in solid phase, due to the interface thickness was negligible. A parameter used in analytical solute trapping model was determined by isothermal phase-field simulation of Ni-Cu alloy. Its application to Si-As and Si-Bi alloys reproduced results that agree reasonably well with experimental data. A comparison between the three models of solute trapping (Aziz, Sobolev and Galenko [Phys. Rev. E, 76, 031606 (2007] was performed. It resulted in large differences in predicting the solidification velocity for partition-less solidification, indicating the necessity for new and more acute experimental data.
Alfonso, Lester; Zamora, Jose; Cruz, Pedro
2015-04-01
The stochastic approach to coagulation considers the coalescence process going in a system of a finite number of particles enclosed in a finite volume. Within this approach, the full description of the system can be obtained from the solution of the multivariate master equation, which models the evolution of the probability distribution of the state vector for the number of particles of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels and initial conditions is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions obtained for the constant and sum kernels, with an excellent correspondence between the analytical and numerical solutions. In order to increase the speedup of the algorithm, software parallelization techniques with OpenMP standard were used, along with an implementation in order to take advantage of new accelerator technologies. Simulations results show an important speedup of the parallelized algorithms. This study was funded by a grant from Consejo Nacional de Ciencia y Tecnologia de Mexico SEP-CONACYT CB-131879. The authors also thanks LUFAC® Computacion SA de CV for CPU time and all the support provided.
A New Method to Solve Numeric Solution of Nonlinear Dynamic System
Directory of Open Access Journals (Sweden)
Min Hu
2016-01-01
Full Text Available It is well known that the cubic spline function has advantages of simple forms, good convergence, approximation, and second-order smoothness. A particular class of cubic spline function is constructed and an effective method to solve the numerical solution of nonlinear dynamic system is proposed based on the cubic spline function. Compared with existing methods, this method not only has high approximation precision, but also avoids the Runge phenomenon. The error analysis of several methods is given via two numeric examples, which turned out that the proposed method is a much more feasible tool applied to the engineering practice.
International Nuclear Information System (INIS)
Houfek, Karel
2008-01-01
Numerical solution of coupled radial differential equations which are encountered in multichannel scattering problems is presented. Numerical approach is based on the combination of the exterior complex scaling method and the finite-elements method with the discrete variable representation. This method can be used not only to solve multichannel scattering problem but also to find bound states and resonance positions and widths directly by diagonalization of the corresponding complex scaled Hamiltonian. Efficiency and accuracy of this method is demonstrated on an analytically solvable two-channel problem.
Keslerová, Radka; Trdlička, David
2015-09-01
This work deals with the numerical modelling of steady flows of incompressible viscous and viscoelastic fluids through the three dimensional channel with T-junction. The fundamental system of equations is the system of generalized Navier-Stokes equations for incompressible fluids. This system is based on the system of balance laws of mass and momentum for incompressible fluids. Two different mathematical models for the stress tensor are used for simulation of Newtonian and Oldroyd-B fluids flow. Numerical solution of the described models is based on cetral finite volume method using explicit Runge-Kutta time integration.
International Nuclear Information System (INIS)
Garratt, T.J.
1989-05-01
Compartment models for the transport of radionuclides in the biosphere are conventionally solved using a numerical time-stepping procedure. This report examines an alternative method based on the numerical inversion of Laplace transforms, which is potentially more efficient and accurate for some classes of problem. The central problem considered is the most efficient and robust technique for solving the Laplace-transformed rate equations. The conclusion is that Gaussian elimination is the most efficient and robust solution method. A general compartment model has been implemented on a personal computer and used to solve a realistic case including radionuclide decay chains. (author)
Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye
2018-04-01
The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.
New numerical method for iterative or perturbative solution of quantum field theory
International Nuclear Information System (INIS)
Hahn, S.C.; Guralnik, G.S.
1999-01-01
A new computational idea for continuum quantum Field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating fermions on a relatively symmetric footing with bosons. As a spin-off of the technology developed for 'exact' solutions, the numerical methods used have a special case application to perturbation theory. We are in the process of developing an entirely numerical approach to evaluating graphs to high perturbative order. (authors)
A numerical solution to the radial equation of the tidal wave propagation
International Nuclear Information System (INIS)
Makarious, S.H.
1981-08-01
The tidal wave function y(x) is a solution to an inhomogeneous, linear, second-order differential equation with variable coefficient. Numerical values for the height-dependence terms, in the observed tides, have been utilized in finding y(x) as a solution to an initial-value problem. Complex Fast Fourier Transform technique is also used to obtain the solution in a complex form. Based on a realistic temperature structure, the atmosphere - below 110 km - has been divided into layers with distinct characteristics, and thus the technique of propagation in stratified media has been applied. The reduced homogeneous equation assumes the form of Helmholtz equation and with initial conditions the general solution is obtained. (author)
Numerical Modeling Tools for the Prediction of Solution Migration Applicable to Mining Site
International Nuclear Information System (INIS)
Martell, M.; Vaughn, P.
1999-01-01
Mining has always had an important influence on cultures and traditions of communities around the globe and throughout history. Today, because mining legislation places heavy emphasis on environmental protection, there is great interest in having a comprehensive understanding of ancient mining and mining sites. Multi-disciplinary approaches (i.e., Pb isotopes as tracers) are being used to explore the distribution of metals in natural environments. Another successful approach is to model solution migration numerically. A proven method to simulate solution migration in natural rock salt has been applied to project through time for 10,000 years the system performance and solution concentrations surrounding a proposed nuclear waste repository. This capability is readily adaptable to simulate solution migration around mining
New numerical solutions of three-dimensional compressible hydrodynamic convection. [in stars
Hossain, Murshed; Mullan, D. J.
1990-01-01
Numerical solutions of three-dimensional compressible hydrodynamics (including sound waves) in a stratified medium with open boundaries are presented. Convergent/divergent points play a controlling role in the flows, which are dominated by a single frequency related to the mean sound crossing time. Superposed on these rapid compressive flows, slower eddy-like flows eventually create convective transport. The solutions contain small structures stacked on top of larger ones, with vertical scales equal to the local pressure scale heights, H sub p. Although convective transport starts later in the evolution, vertical scales of H sub p are apparently selected at much earlier times by nonlinear compressive effects.
Numerical Solutions for Supersonic Flow of an Ideal Gas Around Blunt Two-Dimensional Bodies
Fuller, Franklyn B.
1961-01-01
The method described is an inverse one; the shock shape is chosen and the solution proceeds downstream to a body. Bodies blunter than circular cylinders are readily accessible, and any adiabatic index can be chosen. The lower limit to the free-stream Mach number available in any case is determined by the extent of the subsonic field, which in turn depends upon the body shape. Some discussion of the stability of the numerical processes is given. A set of solutions for flows about circular cylinders at several Mach numbers and several values of the adiabatic index is included.
Determination of Solution Accuracy of Numerical Schemes as Part of Code and Calculation Verification
Energy Technology Data Exchange (ETDEWEB)
Blottner, F.G.; Lopez, A.R.
1998-10-01
This investigation is concerned with the accuracy of numerical schemes for solving partial differential equations used in science and engineering simulation codes. Richardson extrapolation methods for steady and unsteady problems with structured meshes are presented as part of the verification procedure to determine code and calculation accuracy. The local truncation error de- termination of a numerical difference scheme is shown to be a significant component of the veri- fication procedure as it determines the consistency of the numerical scheme, the order of the numerical scheme, and the restrictions on the mesh variation with a non-uniform mesh. Genera- tion of a series of co-located, refined meshes with the appropriate variation of mesh cell size is in- vestigated and is another important component of the verification procedure. The importance of mesh refinement studies is shown to be more significant than just a procedure to determine solu- tion accuracy. It is suggested that mesh refinement techniques can be developed to determine con- sistency of numerical schemes and to determine if governing equations are well posed. The present investigation provides further insight into the conditions and procedures required to effec- tively use Richardson extrapolation with mesh refinement studies to achieve confidence that sim- ulation codes are producing accurate numerical solutions.
Lötstedt, Erik; Jentschura, Ulrich D
2009-02-01
In the relativistic and the nonrelativistic theoretical treatment of moderate and high-power laser-matter interaction, the generalized Bessel function occurs naturally when a Schrödinger-Volkov and Dirac-Volkov solution is expanded into plane waves. For the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field, it is often necessary to evaluate large arrays of generalized Bessel functions, of arbitrary index but with fixed arguments. We show that the generalized Bessel function can be evaluated, in a numerically stable way, by utilizing a recurrence relation and a normalization condition only, without having to compute any initial value. We demonstrate the utility of the method by illustrating the quantum-classical correspondence of the Dirac-Volkov solutions via numerical calculations.
Hassan Fathabadi
2013-01-01
In this study, several novel numerical solutions are presented to solve the turbulent filtration equation and its special case called “Non-Newtonian mechanical filtration equation”. The turbulent filtration equation in porous media is a very important equation which has many applications to solve the problems appearing especially in mechatronics, micro mechanic and fluid mechanic. Many applied mechanical problems can be solved using this equation. For example, non-Newtonian mechanical filtrat...
An efficient approach to the numerical solution of rate-independent problems with nonconvex energies
Czech Academy of Sciences Publication Activity Database
Bartels, S.; Kružík, Martin
2011-01-01
Roč. 9, č. 3 (2011), s. 1275-1300 ISSN 1540-3459 R&D Projects: GA AV ČR IAA100750802 Grant - others:GA ČR(CZ) GAP201/10/0357 Institutional research plan: CEZ:AV0Z10750506 Keywords : numerical solution * nonconvexity Subject RIV: BA - General Mathematics Impact factor: 2.009, year: 2011 http://library.utia.cas.cz/separaty/2011/MTR/kruzik-0364707.pdf
Numerical solution of kinetics equation for point defects accumulation in metals under irradiation
International Nuclear Information System (INIS)
Aldzhambekova, G.T.; Iskakov, B.M.
1999-01-01
In the report the mathematical model, describing processes of generation and accumulation of defects in solids under irradiation is considered. The equations of this model take into account the velocity of Frenkel pairs generation, the mutual recombination of vacancies and the interstitials, as well as velocity of defects absorption by discharge channeling of vacancies and interstitials. By Runge-Kutta method the numerical solution of the model was carried out
The Navier-Stokes-Fourier system: From weak solutions to numerical analysis
Czech Academy of Sciences Publication Activity Database
Feireisl, Eduard
2015-01-01
Roč. 35, č. 3 (2015), s. 185-193 ISSN 0174-4747 R&D Projects: GA ČR GA13-00522S Institutional support: RVO:67985840 Keywords : Navier-Stokes-Fourier system * weak solution * mixed finite-volume finite-element numerical scheme Subject RIV: BA - General Mathematics http://www.degruyter.com/view/j/anly.2015.35.issue-3/anly-2014-1300/anly-2014-1300. xml
Optimality conditions for the numerical solution of optimization problems with PDE constraints :
Energy Technology Data Exchange (ETDEWEB)
Aguilo Valentin, Miguel Alejandro; Ridzal, Denis
2014-03-01
A theoretical framework for the numerical solution of partial di erential equation (PDE) constrained optimization problems is presented in this report. This theoretical framework embodies the fundamental infrastructure required to e ciently implement and solve this class of problems. Detail derivations of the optimality conditions required to accurately solve several parameter identi cation and optimal control problems are also provided in this report. This will allow the reader to further understand how the theoretical abstraction presented in this report translates to the application.
Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic
International Nuclear Information System (INIS)
Douglas, Michael R.; Karp, Robert L.; Lukic, Sergio; Reinbacher, Rene
2007-01-01
We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P 2 . In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic
Numerical Approximations to the Solution of Ray Tracing through the Crystalline Lens
International Nuclear Information System (INIS)
Yildirim, A.; Gökdoğan, A.; Merdan, M.; Lakshminarayanan, V.
2012-01-01
An approximate analytical solution in the form of a rapidly convergent series for tracing light rays through an inhomogeneous graded index medium is developed, using the multi-step differential transform method based on the classical differential transformation method. Numerical results are compared to those obtained by the fourth-order Runge—Kutta method to illustrate the precision and effectiveness of the proposed method. Results are given in explicit and graphical forms. (fundamental areas of phenomenology(including applications))
Lee, Yang-Sub
A time-domain numerical algorithm for solving the KZK (Khokhlov-Zabolotskaya-Kuznetsov) nonlinear parabolic wave equation is developed for pulsed, axisymmetric, finite amplitude sound beams in thermoviscous fluids. The KZK equation accounts for the combined effects of diffraction, absorption, and nonlinearity at the same order of approximation. The accuracy of the algorithm is established via comparison with analytical solutions for several limiting cases, and with numerical results obtained from a widely used algorithm for solving the KZK equation in the frequency domain. The time domain algorithm is used to investigate waveform distortion and shock formation in directive sound beams radiated by pulsed circular piston sources. New results include predictions for the entire process of self-demodulation, and for the effect of frequency modulation on pulse envelope distortion. Numerical results are compared with measurements, and focused sources are investigated briefly.
A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation
Directory of Open Access Journals (Sweden)
Hakon A. Hoel
2007-07-01
Full Text Available We consider a numerical scheme for entropy weak solutions of the DP (Degasperis-Procesi equation $u_t - u_{xxt} + 4uu_x = 3u_{x}u_{xx}+ uu_{xxx}$. Multi-shockpeakons, functions of the form $$ u(x,t =sum_{i=1}^n(m_i(t -hbox{sign}(x-x_i(ts_i(te^{-|x-x_i(t|}, $$ are solutions of the DP equation with a special property; their evolution in time is described by a dynamical system of ODEs. This property makes multi-shockpeakons relatively easy to simulate numerically. We prove that if we are given a non-negative initial function $u_0 in L^1(mathbb{R}cap BV(mathbb{R}$ such that $u_{0} - u_{0,x}$ is a positive Radon measure, then one can construct a sequence of multi-shockpeakons which converges to the unique entropy weak solution in $mathbb{R}imes[0,T$ for any $T>0$. From this convergence result, we construct a multi-shockpeakon based numerical scheme for solving the DP equation.
An analytic solution for numerical modeling validation in electromagnetics: the resistive sphere
Swidinsky, Andrei; Liu, Lifei
2017-11-01
We derive the electromagnetic response of a resistive sphere to an electric dipole source buried in a conductive whole space. The solution consists of an infinite series of spherical Bessel functions and associated Legendre polynomials, and follows the well-studied problem of a conductive sphere buried in a resistive whole space in the presence of a magnetic dipole. Our result is particularly useful for controlled-source electromagnetic problems using a grounded electric dipole transmitter and can be used to check numerical methods of calculating the response of resistive targets (such as finite difference, finite volume, finite element and integral equation). While we elect to focus on the resistive sphere in our examples, the expressions in this paper are completely general and allow for arbitrary source frequency, sphere radius, transmitter position, receiver position and sphere/host conductivity contrast so that conductive target responses can also be checked. Commonly used mesh validation techniques consist of comparisons against other numerical codes, but such solutions may not always be reliable or readily available. Alternatively, the response of simple 1-D models can be tested against well-known whole space, half-space and layered earth solutions, but such an approach is inadequate for validating models with curved surfaces. We demonstrate that our theoretical results can be used as a complementary validation tool by comparing analytic electric fields to those calculated through a finite-element analysis; the software implementation of this infinite series solution is made available for direct and immediate application.
Energy Technology Data Exchange (ETDEWEB)
Ojeda Gonzalez, A.; Domingues, M.O.; Mendes, O., E-mail: ojeda.gonzalez.a@gmail.com [Instituto Nacional de Pesquisas Espaciais (INPE), Sao Jose dos Campos, SP (Brazil); Kaibara, M.K. [Universidade Federal Fluminense (GMA/IME/UFF), Niteroi, RJ (Brazil); Prestes, A. [Universidade do Vale do Paraiba (IP and D/UNIVAP), Sao Jose dos Campos, SP (Brazil). Lab. de Fisica e Astronomia
2015-10-15
The Grad-Shafranov equation is a Poisson's equation, i.e., a partial differential equation of elliptic type. The problem is depending on the initial condition and can be treated as a Cauchy problem. Although it is ill-posed or ill-conditioned, it can be integrated numerically. In the integration of the GS equation, singularities with large values of the potential arise after a certain number of integration steps away from the original data line, and a filter should be used. The Grad-Shafranov reconstruction (GSR) technique was developed from 1996 to 2000 for recovering two-dimensional structures in the magnetopause in an ideal MHD formulation. Other works have used the GSR techniques to study magnetic flux ropes in the solar wind and in the magnetotail from a single spacecraft dataset; posteriorly, it was extended to treat measurements from multiple satellites. From Vlasov equation, it is possible to arrive at the GS-equation in function of the normalized vector potential. A general solution is obtained using complex variable theory. A specific solution was chosen as benchmark case to solve numerically the GS equation.We propose some changes in the resolution scheme of the GS equation to improve the solution. The result of each method is compared with the solution proposed by Hau and Sonnerup (J. Geophys. Res. 104(A4), 6899-6917 (1999)). The main improvement found in the GS resolution was the need to filter Bx values at each y value. (author)
Development of numerical solution techniques in the KIKO3D code
International Nuclear Information System (INIS)
Panka, Istvan; Kereszturi, Andras; Hegedus, Csaba
2005-01-01
The paper describes the numerical methods applied in KIKO3D three-dimensional reactor dynamics code and present a new, more effective method (Bi-CGSTAB) for accelerating the large sparse matrix equation solution. The convergence characteristics were investigated in a given macro time step of a Control Rod Ejection transient. The results obtained by the old GMRES and new Bi-CGSTAB methods are compared. It is concluded that the real relative errors of the solutions obtained by GMRES or Bi - CGSTAB algorithms are in fact closer together than the estimated relative errors. The KIKO3D-Bi-CGSTAB method converges safely and it is 7-12 % faster than the old KIKO3D-GMRES solution (Authors)
Solved problems in classical mechanics analytical and numerical solutions with comments
de Lange, O L
2010-01-01
Apart from an introductory chapter giving a brief summary of Newtonian and Lagrangian mechanics, this book consists entirely of questions and solutions on topics in classical mechanics that will be encountered in undergraduate and graduate courses. These include one-, two-, and three- dimensional motion; linear and nonlinear oscillations; energy, potentials, momentum, and angular momentum; spherically symmetric potentials; multi-particle systems; rigid bodies; translation androtation of the reference frame; the relativity principle and some of its consequences. The solutions are followed by a set of comments intended to stimulate inductive reasoning and provide additional information of interest. Both analytical and numerical (computer) techniques are used to obtain andanalyze solutions. The computer calculations use Mathematica (version 7), and the relevant code is given in the text. It includes use of the interactive Manipulate function which enables one to observe simulated motion on a computer screen, and...
Solution of AntiSeepage for Mengxi River Based on Numerical Simulation of Unsaturated Seepage
Ji, Youjun; Zhang, Linzhi; Yue, Jiannan
2014-01-01
Lessening the leakage of surface water can reduce the waste of water resources and ground water pollution. To solve the problem that Mengxi River could not store water enduringly, geology investigation, theoretical analysis, experiment research, and numerical simulation analysis were carried out. Firstly, the seepage mathematical model was established based on unsaturated seepage theory; secondly, the experimental equipment for testing hydraulic conductivity of unsaturated soil was developed to obtain the curve of two-phase flow. The numerical simulation of leakage in natural conditions proves the previous inference and leakage mechanism of river. At last, the seepage control capacities of different impervious materials were compared by numerical simulations. According to the engineering actuality, the impervious material was selected. The impervious measure in this paper has been proved to be effectible by hydrogeological research today. PMID:24707199
Solution of AntiSeepage for Mengxi River Based on Numerical Simulation of Unsaturated Seepage
Directory of Open Access Journals (Sweden)
Youjun Ji
2014-01-01
Full Text Available Lessening the leakage of surface water can reduce the waste of water resources and ground water pollution. To solve the problem that Mengxi River could not store water enduringly, geology investigation, theoretical analysis, experiment research, and numerical simulation analysis were carried out. Firstly, the seepage mathematical model was established based on unsaturated seepage theory; secondly, the experimental equipment for testing hydraulic conductivity of unsaturated soil was developed to obtain the curve of two-phase flow. The numerical simulation of leakage in natural conditions proves the previous inference and leakage mechanism of river. At last, the seepage control capacities of different impervious materials were compared by numerical simulations. According to the engineering actuality, the impervious material was selected. The impervious measure in this paper has been proved to be effectible by hydrogeological research today.
International Nuclear Information System (INIS)
Molinero, J.; Samper, J.
2003-01-01
Several countries around the world are considering the final disposal of high-level radioactive waste in deep repositories located in fractured granite formations. Evaluating the long term safety of such repositories requires sound conceptual and numerical models which must consider simultaneously groundwater flow, solute transport and chemical and radiological processes. These models are being developed from data and knowledge gained from in situ experiments carried out at deep underground laboratories such as that of Aspo, Sweden, constructed in fractured granite. The Redox Zone Experiment is one of such experiments performed at Aspo in order to evaluate the effects of the construction of the access tunnel on the hydrogeological and hydrochemical conditions of a fracture zone intersected by the tunnel. Previous authors interpreted hydrochemical and isotopic data of this experiment using a mass-balance approach based on a qualitative description of groundwater flow conditions. Such an interpretation, however, is subject to uncertainties related to an over-simplified conceptualization of groundwater flow. Here we present numerical models of groundwater flow and solute transport for this fracture zone. The first model is based on previously published conceptual model. It presents noticeable un consistencies and fails to match simultaneously observed draw downs and chloride breakthrough curves. To overcome its limitations, a revised flow and transport model is presented which relies directly on available hydrodynamic and transport parameters, is based on the identification of appropriate flow and transport boundary conditions and uses, when needed, solute data extrapolated from nearby fracture zones. A significant quantitative improvement is achieved with the revised model because its results match simultaneously drawdown and chloride data. Other improvements are qualitative and include: ensuring consistency of hydrodynamic and hydrochemical data and avoiding
International Nuclear Information System (INIS)
Milioli, F.E.
1985-01-01
In this research work a numerical model for the solution of two-dimensional natural convection problems in arbitrary cavities of a Boussinesq fluid is presented. The conservation equations are written in a general curvilinear coordinate system which matches the irregular boundaries of the domain. The nonorthogonal system is generated by a suitable system of elliptic equations. The momentum and continuity equations are transformed from the Cartesian system to the general curvilinear system keeping the Cartesian velocity components as the dependent variables in the transformed domain. Finite difference equations are obtained for the contravariant velocity components in the transformed domain. The numerical calculations are performed in a fixed rectangular domain and both the Cartesian and the contravariant velocity components take part in the solutiomn procedure. The dependent variables are arranged on the grid in a staggered manner. The numerical model is tested by solving the driven flow in a square cavity with a moving side using a nonorthogoanl grid. The natural convenction in a square cavity, using an orthogonal and a nonorthogonal grid, is also solved for the model test. Also, the solution for the buoyancy flow between a square cylinder placed inside a circular cylinder is presented. The results of the test problems are compared with those available in the specialized literature. Finally, in order to show the generality of the model, the natural convection problem inside a very irregular cavity is presented. (Author) [pt
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui
2014-01-01
Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904
Enriched Meshfree Method for an Accurate Numerical Solution of the Motz Problem
Directory of Open Access Journals (Sweden)
Won-Tak Hong
2016-01-01
Full Text Available We present an enriched meshfree solution of the Motz problem. The Motz problem has been known as a benchmark problem to verify the efficiency of numerical methods in the presence of a jump boundary data singularity at a point, where an abrupt change occurs for the boundary condition. We propose a singular basis function enrichment technique in the context of partition of unity based meshfree method. We take the leading terms of the local series expansion at the point singularity and use them as enrichment functions for the local approximation space. As a result, we obtain highly accurate leading coefficients of the Motz problem that are comparable to the most accurate numerical solution. The proposed singular enrichment technique is highly effective in the case of the local series expansion of the solution being known. The enrichment technique that is used in this study can be applied to monotone singularities (of type rα with α<1 as well as oscillating singularities (of type rαsin(ϵlogr. It is the first attempt to apply singular meshfree enrichment technique to the Motz problem.
The numerical solution of thawing process in phase change slab using variable space grid technique
Directory of Open Access Journals (Sweden)
Serttikul, C.
2007-09-01
Full Text Available This paper focuses on the numerical analysis of melting process in phase change material which considers the moving boundary as the main parameter. In this study, pure ice slab and saturated porous packed bed are considered as the phase change material. The formulation of partial differential equations is performed consisting heat conduction equations in each phase and moving boundary equation (Stefan equation. The variable space grid method is then applied to these equations. The transient heat conduction equations and the Stefan condition are solved by using the finite difference method. A one-dimensional melting model is then validated against the available analytical solution. The effect of constant temperature heat source on melting rate and location of melting front at various times is studied in detail.It is found that the nonlinearity of melting rate occurs for a short time. The successful comparison with numerical solution and analytical solution should give confidence in the proposed mathematical treatment, and encourage the acceptance of this method as useful tool for exploring practical problems such as forming materials process, ice melting process, food preservation process and tissue preservation process.
Efficient numerical methods for the large-scale, parallel solution of elastoplastic contact problems
Frohne, Jörg
2015-08-06
© 2016 John Wiley & Sons, Ltd. Quasi-static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid-preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three-dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns.
Efficient numerical methods for the large-scale, parallel solution of elastoplastic contact problems
Frohne, Jö rg; Heister, Timo; Bangerth, Wolfgang
2015-01-01
© 2016 John Wiley & Sons, Ltd. Quasi-static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid-preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three-dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns.
Numerical solution of multi group-Two dimensional- Adjoint equation with finite element method
International Nuclear Information System (INIS)
Poursalehi, N.; Khalafi, H.; Shahriari, M.; Minoochehr
2008-01-01
Adjoint equation is used for perturbation theory in nuclear reactor design. For numerical solution of adjoint equation, usually two methods are applied. These are Finite Element and Finite Difference procedures. Usually Finite Element Procedure is chosen for solving of adjoint equation, because it is more use able in variety of geometries. In this article, Galerkin Finite Element method is discussed. This method is applied for numerical solving multi group, multi region and two dimensional (X, Y) adjoint equation. Typical reactor geometry is partitioned with triangular meshes and boundary condition for adjoint flux is considered zero. Finally, for a case of defined parameters, Finite Element Code was applied and results were compared with Citation Code
Long-time behavior in numerical solutions of certain dynamical systems
International Nuclear Information System (INIS)
Vazquez, L.
1987-01-01
A general discretization of the ordinary nonlinear differential equations d 2 v/dt 2 =f(v) and dv/dt=g(v) is studied. The discrete scheme conserves the discrete analogous of a quantity that is conserved by the corresponding equations. This method is applied to two cases and no ''ghost solutions'' were observed for the long range calculation. In these cases we analyze the stability of the corresponding numerical scheme as a dynamical system and in the sense studied by Kuo Pen-Yu and Stetter. In particular we find a correspondence between both kinds of stability. (author)
The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory
International Nuclear Information System (INIS)
Woznicki, Z.I.
1994-01-01
The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs
Turchi, Patrice E. A.; Fattebert, Jean-Luc; Dorr, Milo R.; Wickett, Michael E.; Belak, James F.
2011-03-01
We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in alloys using physical parameters from thermodynamic (CALPHAD) and kinetic databases. The coupled system of PFM equations includes a local order parameter, a quaternion representation of local crystal orientation and a species composition parameter. Time evolution of microstructures and alloy composition is obtained using an implicit time integration of the system. Physical parameters in databases can be obtained either through experiment or first-principles calculations. Application to coring studies and microstructure evolution of Au-Ni will be presented. Prepared by LLNL under Contract DE-AC52-07NA27344
Human-computer interfaces applied to numerical solution of the Plateau problem
Elias Fabris, Antonio; Soares Bandeira, Ivana; Ramos Batista, Valério
2015-09-01
In this work we present a code in Matlab to solve the Problem of Plateau numerically, and the code will include human-computer interface. The Problem of Plateau has applications in areas of knowledge like, for instance, Computer Graphics. The solution method will be the same one of the Surface Evolver, but the difference will be a complete graphical interface with the user. This will enable us to implement other kinds of interface like ocular mouse, voice, touch, etc. To date, Evolver does not include any graphical interface, which restricts its use by the scientific community. Specially, its use is practically impossible for most of the Physically Challenged People.
Born approximation to a perturbative numerical method for the solution of the Schrodinger equation
International Nuclear Information System (INIS)
Adam, Gh.
1978-05-01
A perturbative numerical (PN) method is given for the solution of a regular one-dimensional Cauchy problem arising from the Schroedinger equation. The present method uses a step function approximation for the potential. Global, free of scaling difficulty, forward and backward PN algorithms are derived within first order perturbation theory (Born approximation). A rigorous analysis of the local truncation errors is performed. This shows that the order of accuracy of the method is equal to four. In between the mesh points, the global formula for the wavefunction is accurate within O(h 4 ), while that for the first order derivative is accurate within O(h 3 ). (author)
Numerical solution of the Schrodinger equation for stationary bound states using nodel theorem
International Nuclear Information System (INIS)
Chen Zhijiang; Kong Fanmei; Din Yibin
1987-01-01
An iterative procedure for getting the numerical solution of Schrodinger equation on stationary bound states is introduced. The theoretical foundtion, the practical steps and the method are presented. An example is added at the end. Comparing with other methods, the present one requires less storage, less running time but posesses higher accuracy. It can be run on the personal computer or microcomputer with 256 K memory and 16 bit word length such as IBM/PC, MC68000/83/20, PDP11/23 etc
International Nuclear Information System (INIS)
Carver, M.B.
1995-08-01
The discussion briefly establishes some requisite concepts of differential equation theory, and applies these to describe methods for numerical solution of the thermalhydraulic conservation equations in their various forms. The intent is to cover the general methodology without obscuring the principles with details. As a short overview of computational thermalhydraulics, the material provides an introductory foundation, so that those working on the application of thermalhydraulic codes can begin to understand the many intricacies involved without having to locate and read the references given. Those intending to work in code development will need to read and understand all the references. (author). 49 refs
The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory
Energy Technology Data Exchange (ETDEWEB)
Woznicki, Z I [Institute of Atomic Energy, Otwock-Swierk (Poland)
1994-12-31
The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs.
Rosenbaum, J. S.
1971-01-01
Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.
The numerical analysis of eigenvalue problem solutions in multigroup neutron diffusion theory
International Nuclear Information System (INIS)
Woznicki, Z.I.
1995-01-01
The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iterations within global iterations. Particular iterative strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 35 figs, 16 tabs
The numerical solution of linear multi-term fractional differential equations: systems of equations
Edwards, John T.; Ford, Neville J.; Simpson, A. Charles
2002-11-01
In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.
WATSFAR: numerical simulation of soil WATer and Solute fluxes using a FAst and Robust method
Crevoisier, David; Voltz, Marc
2013-04-01
To simulate the evolution of hydro- and agro-systems, numerous spatialised models are based on a multi-local approach and improvement of simulation accuracy by data-assimilation techniques are now used in many application field. The latest acquisition techniques provide a large amount of experimental data, which increase the efficiency of parameters estimation and inverse modelling approaches. In turn simulations are often run on large temporal and spatial domains which requires a large number of model runs. Eventually, despite the regular increase in computing capacities, the development of fast and robust methods describing the evolution of saturated-unsaturated soil water and solute fluxes is still a challenge. Ross (2003, Agron J; 95:1352-1361) proposed a method, solving 1D Richards' and convection-diffusion equation, that fulfil these characteristics. The method is based on a non iterative approach which reduces the numerical divergence risks and allows the use of coarser spatial and temporal discretisations, while assuring a satisfying accuracy of the results. Crevoisier et al. (2009, Adv Wat Res; 32:936-947) proposed some technical improvements and validated this method on a wider range of agro- pedo- climatic situations. In this poster, we present the simulation code WATSFAR which generalises the Ross method to other mathematical representations of soil water retention curve (i.e. standard and modified van Genuchten model) and includes a dual permeability context (preferential fluxes) for both water and solute transfers. The situations tested are those known to be the less favourable when using standard numerical methods: fine textured and extremely dry soils, intense rainfall and solute fluxes, soils near saturation, ... The results of WATSFAR have been compared with the standard finite element model Hydrus. The analysis of these comparisons highlights two main advantages for WATSFAR, i) robustness: even on fine textured soil or high water and solute
Czech Academy of Sciences Publication Activity Database
Papež, Jan; Liesen, J.; Strakoš, Z.
2014-01-01
Roč. 449, 15 May (2014), s. 89-114 ISSN 0024-3795 R&D Projects: GA AV ČR IAA100300802; GA ČR GA201/09/0917 Grant - others:GA MŠk(CZ) LL1202; GA UK(CZ) 695612 Institutional support: RVO:67985807 Keywords : numerical solution of partial differential equations * finite element method * adaptivity * a posteriori error analysis * discretization error * algebra ic error * spatial distribution of the error Subject RIV: BA - General Mathematics Impact factor: 0.939, year: 2014
Voytishek, Anton V.; Shipilov, Nikolay M.
2017-11-01
In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms.
Numerical Methods for Solution of the Extended Linear Quadratic Control Problem
DEFF Research Database (Denmark)
Jørgensen, John Bagterp; Frison, Gianluca; Gade-Nielsen, Nicolai Fog
2012-01-01
In this paper we present the extended linear quadratic control problem, its efficient solution, and a discussion of how it arises in the numerical solution of nonlinear model predictive control problems. The extended linear quadratic control problem is the optimal control problem corresponding...... to the Karush-Kuhn-Tucker system that constitute the majority of computational work in constrained nonlinear and linear model predictive control problems solved by efficient MPC-tailored interior-point and active-set algorithms. We state various methods of solving the extended linear quadratic control problem...... and discuss instances in which it arises. The methods discussed in the paper have been implemented in efficient C code for both CPUs and GPUs for a number of test examples....
Numerical modelling of coupled fluid, heat, and solute transport in deformable fractured rock
International Nuclear Information System (INIS)
Chan, T.; Reid, J.A.K.
1987-01-01
This paper reports on a three-dimensional (3D) finite-element code, MOTIF (model of transport in fractured/porous media), developed to model the coupled processes of groundwater flow, heat transport, brine transport, and one-species radionuclide transport in geological media. Three types of elements are available: a 3D continuum element, a planar fracture element that can be oriented in any arbitrary direction in 3D space or pipe flow in 3D space, and a line element for simulating fracture flow in 2D space or pipe flow in 3D space. As a quality-assurance measure, the MOTIF code was verified by comparison of its results with analytical solutions and other published numerical solutions
International Nuclear Information System (INIS)
Graf, U.
1986-01-01
A combination of several numerical methods is used to construct a procedure for effective calculation of complex three-dimensional fluid flow problems. The split coefficient matrix (SCM) method is used so that the differenced equations of the hyperbolic system do not disturb correct signal propagation. The semi-discretisation of the equations of the SCM method is done with the asymmetric, separated region, weighted residual (ASWR) method to give accurate solutions on a relatively coarse mesh. For the resulting system of ordinary differential equations, a general-purpose ordinary differential equation solver is used in conjunction with a method of fractional steps for an economic solution of the large system of linear equations. (orig.) [de
Numerical solution of quadratic matrix equations for free vibration analysis of structures
Gupta, K. K.
1975-01-01
This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.
Energy Technology Data Exchange (ETDEWEB)
Liu, L.H. [School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001 (China)]. E-mail: lhliu@hit.edu.cn
2006-11-15
In graded index media, the ray goes along a curved path determined by Fermat principle. Generally, the curved ray trajectory in graded index media is a complex implicit function, and the curved ray tracing is very difficult and complex. Only for some special refractive index distributions, the curved ray trajectory can be expressed as a simple explicit function. Two important examples are the layered and the radial graded index distributions. In this paper, the radiative heat transfer problems in two-dimensional square semitransparent with layered and radial graded index distributions are analyzed. After deduction of the ray trajectory, the radiative heat transfer problems are solved by using the Monte Carlo curved ray-tracing method. Some numerical solutions of dimensionless net radiative heat flux and medium temperature are tabulated as the benchmark solutions for the future development of approximation techniques for multi-dimensional radiative heat transfer in graded index media.
Numerical Simulation of the Freeze-Thaw Behavior of Mortar Containing Deicing Salt Solution.
Esmaeeli, Hadi S; Farnam, Yaghoob; Bentz, Dale P; Zavattieri, Pablo D; Weiss, Jason
2017-02-01
This paper presents a one-dimensional finite difference model that is developed to describe the freeze-thaw behavior of an air-entrained mortar containing deicing salt solution. A phenomenological model is used to predict the temperature and the heat flow for mortar specimens during cooling and heating. Phase transformations associated with the freezing/melting of water/ice or transition of the eutectic solution from liquid to solid are included in this phenomenological model. The lever rule is used to calculate the quantity of solution that undergoes the phase transformation, thereby simulating the energy released/absorbed during phase transformation. Undercooling and pore size effects are considered in the numerical model. To investigate the effect of pore size distribution, this distribution is considered using the Gibbs-Thomson equation in a saturated mortar specimen. For an air-entrained mortar, the impact of considering pore size (and curvature) on freezing was relatively insignificant; however the impact of pore size is much more significant during melting. The fluid inside pores smaller than 5 nm (i.e., gel pores) has a relatively small contribution in the macroscopic freeze-thaw behavior of mortar specimens within the temperature range used in this study (i.e., +24 °C to -35 °C), and can therefore be neglected for the macroscopic freeze-thaw simulations. A heat sink term is utilized to simulate the heat dissipation during phase transformations. Data from experiments performed using a low-temperature longitudinal guarded comparative calorimeter (LGCC) on mortar specimens fully saturated with various concentration NaCl solutions or partially saturated with water is compared to the numerical results and a promising agreement is generally obtained.
International Nuclear Information System (INIS)
Saha Ray, S.; Patra, A.
2012-01-01
Highlights: ► In this paper fractional neutron point kinetic equation has been analyzed. ► The numerical solution for fractional neutron point kinetic equation is obtained. ► Explicit Finite Difference Method has been applied. ► Supercritical reactivity, critical reactivity and subcritical reactivity analyzed. ► Comparison between fractional and classical neutron density is presented. - Abstract: In the present article, a numerical procedure to efficiently calculate the solution for fractional point kinetics equation in nuclear reactor dynamics is investigated. The Explicit Finite Difference Method is applied to solve the fractional neutron point kinetic equation with the Grunwald–Letnikov (GL) definition (). Fractional Neutron Point Kinetic Model has been analyzed for the dynamic behavior of the neutron motion in which the relaxation time associated with a variation in the neutron flux involves a fractional order acting as exponent of the relaxation time, to obtain the best operation of a nuclear reactor dynamics. Results for neutron dynamic behavior for subcritical reactivity, supercritical reactivity and critical reactivity and also for different values of fractional order have been presented and compared with the classical neutron point kinetic (NPK) equation as well as the results obtained by the learned researchers .
Underestimation of nuclear fuel burnup – theory, demonstration and solution in numerical models
Directory of Open Access Journals (Sweden)
Gajda Paweł
2016-01-01
Full Text Available Monte Carlo methodology provides reference statistical solution of neutron transport criticality problems of nuclear systems. Estimated reaction rates can be applied as an input to Bateman equations that govern isotopic evolution of reactor materials. Because statistical solution of Boltzmann equation is computationally expensive, it is in practice applied to time steps of limited length. In this paper we show that simple staircase step model leads to underprediction of numerical fuel burnup (Fissions per Initial Metal Atom – FIMA. Theoretical considerations indicates that this error is inversely proportional to the length of the time step and origins from the variation of heating per source neutron. The bias can be diminished by application of predictor-corrector step model. A set of burnup simulations with various step length and coupling schemes has been performed. SERPENT code version 1.17 has been applied to the model of a typical fuel assembly from Pressurized Water Reactor. In reference case FIMA reaches 6.24% that is equivalent to about 60 GWD/tHM of industrial burnup. The discrepancies up to 1% have been observed depending on time step model and theoretical predictions are consistent with numerical results. Conclusions presented in this paper are important for research and development concerning nuclear fuel cycle also in the context of Gen4 systems.
Numerical modeling of solute transport in deformable unsaturated layered soil
Directory of Open Access Journals (Sweden)
Sheng Wu
2017-07-01
Full Text Available The effect of soil stratification was studied through numerical investigation based on the coupled model of solute transport in deformable unsaturated soil. The theoretical model implied two-way coupled excess pore pressure and soil deformation based on Biot's consolidation theory as well as a one-way coupled volatile pollutant concentration field developed from the advection-diffusion theory. Embedded in the model, the degree of saturation, fluid compressibility, self-weight of the soil matrix, porosity variance, longitudinal dispersion, and linear sorption were computed. Based on simulation results of a proposed three-layer landfill model using the finite element method, the multi-layer effects are discussed with regard to the hydraulic conductivity, shear modulus, degree of saturation, molecular diffusion coefficient, and thickness of each layer. Generally speaking, contaminants spread faster in a stratified field with a soft and highly permeable top layer; soil parameters of the top layer are more critical than the lower layers but controlling soil thicknesses will alter the results. This numerical investigation showed noticeable impacts of stratified soil properties on solute migration results, demonstrating the importance of correctly modeling layered soil instead of simply assuming the averaged properties across the soil profile.
Fikri, Fariz Fahmi; Nuraini, Nuning
2018-03-01
The differential equation is one of the branches in mathematics which is closely related to human life problems. Some problems that occur in our life can be modeled into differential equations as well as systems of differential equations such as the Lotka-Volterra model and SIR model. Therefore, solving a problem of differential equations is very important. Some differential equations are difficult to solve, so numerical methods are needed to solve that problems. Some numerical methods for solving differential equations that have been widely used are Euler Method, Heun Method, Runge-Kutta and others. However, some of these methods still have some restrictions that cause the method cannot be used to solve more complex problems such as an evaluation interval that we cannot change freely. New methods are needed to improve that problems. One of the method that can be used is the artificial bees colony algorithm. This algorithm is one of metaheuristic algorithm method, which can come out from local search space and do exploration in solution search space so that will get better solution than other method.
Numerical solution of non-linear dual-phase-lag bioheat transfer equation within skin tissues.
Kumar, Dinesh; Kumar, P; Rai, K N
2017-11-01
This paper deals with numerical modeling and simulation of heat transfer in skin tissues using non-linear dual-phase-lag (DPL) bioheat transfer model under periodic heat flux boundary condition. The blood perfusion is assumed temperature-dependent which results in non-linear DPL bioheat transfer model in order to predict more accurate results. A numerical method of line which is based on finite difference and Runge-Kutta (4,5) schemes, is used to solve the present non-linear problem. Under specific case, the exact solution has been obtained and compared with the present numerical scheme, and we found that those are in good agreement. A comparison based on model selection criterion (AIC) has been made among non-linear DPL models when the variation of blood perfusion rate with temperature is of constant, linear and exponential type with the experimental data and it has been found that non-linear DPL model with exponential variation of blood perfusion rate is closest to the experimental data. In addition, it is found that due to absence of phase-lag phenomena in Pennes bioheat transfer model, it achieves steady state more quickly and always predict higher temperature than thermal and DPL non-linear models. The effect of coefficient of blood perfusion rate, dimensionless heating frequency and Kirchoff number on dimensionless temperature distribution has also been analyzed. The whole analysis is presented in dimensionless form. Copyright © 2017 Elsevier Inc. All rights reserved.
International Nuclear Information System (INIS)
Fenwick, John D.; Pardo-Montero, Juan
2010-01-01
Purpose: Homogenized blocked arcs are intuitively appealing as basis functions for multicriteria optimization of rotational radiotherapy. Such arcs avoid an organ-at-risk (OAR), spread dose out well over the rest-of-body (ROB), and deliver homogeneous doses to a planning target volume (PTV) using intensity modulated fluence profiles, obtainable either from closed-form solutions or iterative numerical calculations. Here, the analytic and iterative arcs are compared. Methods: Dose-distributions have been calculated for nondivergent beams, both including and excluding scatter, beam penumbra, and attenuation effects, which are left out of the derivation of the analytic arcs. The most straightforward analytic arc is created by truncating the well-known Brahme, Roos, and Lax (BRL) solution, cutting its uniform dose region down from an annulus to a smaller nonconcave region lying beyond the OAR. However, the truncation leaves behind high dose hot-spots immediately on either side of the OAR, generated by very high BRL fluence levels just beyond the OAR. These hot-spots can be eliminated using alternative analytical solutions ''C'' and ''L,'' which, respectively, deliver constant and linearly rising fluences in the gap region between the OAR and PTV (before truncation). Results: Measured in terms of PTV dose homogeneity, ROB dose-spread, and OAR avoidance, C solutions generate better arc dose-distributions than L when scatter, penumbra, and attenuation are left out of the dose modeling. Including these factors, L becomes the best analytical solution. However, the iterative approach generates better dose-distributions than any of the analytical solutions because it can account and compensate for penumbra and scatter effects. Using the analytical solutions as starting points for the iterative methodology, dose-distributions almost as good as those obtained using the conventional iterative approach can be calculated very rapidly. Conclusions: The iterative methodology is
Sun, Shuyu; Salama, Amgad; El-Amin, Mohamed
2012-01-01
In this paper we introduce a new technique for the numerical solution of the various partial differential equations governing flow and transport phenomena in porous media. This method is proposed to be used in high level programming languages like
Lucas, P.; Van Zuijlen, A.H.; Bijl, H.
2009-01-01
Mesh adaptation is a fairly established tool to obtain numerically accurate solutions for flow problems. Computational efficiency is, however, not always guaranteed for the adaptation strategies found in literature. Typically excessive mesh growth diminishes the potential efficiency gain. This
Directory of Open Access Journals (Sweden)
Vanessa Cuenca
2016-12-01
Full Text Available This paper describes experimental formulations of polymeric solutions through lab evaluations with the objective of finding optimum solution concentration to fluid mobility in reservoirs as previous step before implementing a pilot project of enhanced oil recovery. The polymers, firstly, were selected based on the properties from fluids from reservoir. Two types of polymers were used TCC-330 and EOR909 and the experimental tests were: thermal stability, compatibility, adsorption, salinity, and displacement. The design with the best results was with polymer TCC-330 at 1,500 ppm concentration.
Numerical solution of chemically reactive non-Newtonian fluid flow: Dual stratification
Rehman, Khalil Ur; Malik, M. Y.; Khan, Abid Ali; Zehra, Iffat; Zahri, Mostafa; Tahir, M.
2017-12-01
We have found that only a few attempts are available in the literature relatively to the tangent hyperbolic fluid flow induced by stretching cylindrical surfaces. In particular, temperature and concentration stratification effects have not been investigated until now with respect to the tangent hyperbolic fluid model. Therefore, we have considered the tangent hyperbolic fluid flow induced by an acutely inclined cylindrical surface in the presence of both temperature and concentration stratification effects. To be more specific, the fluid flow is attained with the no slip condition, which implies that the bulk motion of the fluid particles is the same as the stretching velocity of a cylindrical surface. Additionally, the flow field situation is manifested with heat generation, mixed convection and chemical reaction effects. The flow partial differential equations give a complete description of the present problem. Therefore, to trace out the solution, a set of suitable transformations is introduced to convert these equations into ordinary differential equations. In addition, a self-coded computational algorithm is executed to inspect the numerical solution of these reduced equations. The effect logs of the involved parameters are provided graphically. Furthermore, the variations of the physical quantities are examined and given with the aid of tables. It is observed that the fluid temperature is a decreasing function of the thermal stratification parameter and a similar trend is noticed for the concentration via the solutal stratification parameter.
Boundary integral equation methods and numerical solutions thin plates on an elastic foundation
Constanda, Christian; Hamill, William
2016-01-01
This book presents and explains a general, efficient, and elegant method for solving the Dirichlet, Neumann, and Robin boundary value problems for the extensional deformation of a thin plate on an elastic foundation. The solutions of these problems are obtained both analytically—by means of direct and indirect boundary integral equation methods (BIEMs)—and numerically, through the application of a boundary element technique. The text discusses the methodology for constructing a BIEM, deriving all the attending mathematical properties with full rigor. The model investigated in the book can serve as a template for the study of any linear elliptic two-dimensional problem with constant coefficients. The representation of the solution in terms of single-layer and double-layer potentials is pivotal in the development of a BIEM, which, in turn, forms the basis for the second part of the book, where approximate solutions are computed with a high degree of accuracy. The book is intended for graduate students and r...
Numerical solution of large nonlinear boundary value problems by quadratic minimization techniques
International Nuclear Information System (INIS)
Glowinski, R.; Le Tallec, P.
1984-01-01
The objective of this paper is to describe the numerical treatment of large highly nonlinear two or three dimensional boundary value problems by quadratic minimization techniques. In all the different situations where these techniques were applied, the methodology remains the same and is organized as follows: 1) derive a variational formulation of the original boundary value problem, and approximate it by Galerkin methods; 2) transform this variational formulation into a quadratic minimization problem (least squares methods) or into a sequence of quadratic minimization problems (augmented lagrangian decomposition); 3) solve each quadratic minimization problem by a conjugate gradient method with preconditioning, the preconditioning matrix being sparse, positive definite, and fixed once for all in the iterative process. This paper will illustrate the methodology above on two different examples: the description of least squares solution methods and their application to the solution of the unsteady Navier-Stokes equations for incompressible viscous fluids; the description of augmented lagrangian decomposition techniques and their application to the solution of equilibrium problems in finite elasticity
Baehr, Arthur L.; Corapcioglu, M. Yavuz
1987-01-01
In this paper we develop a numerical solution to equations developed in part 1 (M. Y. Corapcioglu and A. L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method of forward projection to evaluate nonlinear coefficients, provides estimates of the flux of solubilized hydrocarbon constituents to groundwater from the portion of a spill which remains trapped in a soil after routine remedial efforts to recover the product have ceased. The procedure was used to solve the one-dimensional (vertical) form of the system of nonlinear partial differential equations defining the transport for each constituent of the product. Additionally, a homogeneous, isothermal soil with constant water content was assumed. An equilibrium assumption partitions the constituents between air, water, adsorbed, and immiscible phases. Free oxygen transport in the soil was also simulated to provide an upper bound estimate of aerobic biodgradation rates. Results are presented for a hypothetical gasoline consisting of eight groups of hydrocarbon constituents. Rates at which hydrocarbon mass is removed from the soil, entering either the atmosphere or groundwater, or is biodegraded are presented. A significant sensitivity to model parameters, particularly the parameters characterizing diffusive vapor transport, was discovered. We conclude that hydrocarbon solute composition in groundwater beneath a gasoline contaminated soil would be heavily weighted toward aromatic constituents like benzene, toluene, and xylene.
Directory of Open Access Journals (Sweden)
M. Boumaza
2015-07-01
Full Text Available Transient convection heat transfer is of fundamental interest in many industrial and environmental situations, as well as in electronic devices and security of energy systems. Transient fluid flow problems are among the more difficult to analyze and yet are very often encountered in modern day technology. The main objective of this research project is to carry out a theoretical and numerical analysis of transient convective heat transfer in vertical flows, when the thermal field is due to different kinds of variation, in time and space of some boundary conditions, such as wall temperature or wall heat flux. This is achieved by the development of a mathematical model and its resolution by suitable numerical methods, as well as performing various sensitivity analyses. These objectives are achieved through a theoretical investigation of the effects of wall and fluid axial conduction, physical properties and heat capacity of the pipe wall on the transient downward mixed convection in a circular duct experiencing a sudden change in the applied heat flux on the outside surface of a central zone.
Numerical solutions of the monoenergetic neutron transport equation with anisotropic scattering
International Nuclear Information System (INIS)
Dahl, B.
1985-01-01
The Boltzmann equation for monoenergetic neutrons has been solved numerically with high accuracy for homogeneous slabs and spheres with various degree of linear anisotropy. Vacuum boundary conditions are used. The numerical method is based on previous work by Carlvik. Benchmark values of the criticality factor and higher order eigenvalues are given for multiplying systems of thickness or diameter from 10 -5 to 20 mean free paths and with anisotropy coefficients from 0.0 to 0.3. For slab geometry, both even and odd mode eigenvalues are treated. With increasing anisotropy, an increasing number of complex eigenvalues is observer. The total flux is calculated from the eigenvector and tables of the fundamental mode flux are given. Accurate extrapolation distances are derived for various dimensions and anisotropy coefficients from our eigenvalue results on slabs and spheres and from the work by Sanchez on infinite cylinders.The time eigenvalue spectrum in subcritical systems has also been studied. First, the connection between the eigenvalues arising from the time dependent and stationary transport equation is established. Based on this, the spectrum of real time eigenvalues in slabs and spheres is calculated. For spheres, the existence of complex time eigenvalues in the region beyond the value corresponding to the Corngold limit is numerically established. The presence of such eigenvalues has earlier not been proved. It is further shown that the Boltzmann equation for a sphere is significantly simplified when the decay constant is at the Corngold limit. The spectrum of sphere diameters corresponding to this decay constant is calculated for various linear anisotropies, and detailed numerical results are given. (Author)
Zhang, Chong; Lü, Qingtian; Yan, Jiayong; Qi, Guang
2018-04-01
Downward continuation can enhance small-scale sources and improve resolution. Nevertheless, the common methods have disadvantages in obtaining optimal results because of divergence and instability. We derive the mean-value theorem for potential fields, which could be the theoretical basis of some data processing and interpretation. Based on numerical solutions of the mean-value theorem, we present the convergent and stable downward continuation methods by using the first-order vertical derivatives and their upward continuation. By applying one of our methods to both the synthetic and real cases, we show that our method is stable, convergent and accurate. Meanwhile, compared with the fast Fourier transform Taylor series method and the integrated second vertical derivative Taylor series method, our process has very little boundary effect and is still stable in noise. We find that the characters of the fading anomalies emerge properly in our downward continuation with respect to the original fields at the lower heights.
International Nuclear Information System (INIS)
Kovac, D.; Otto, G.; Hobler, G.
2005-01-01
In this paper we present a model of amorphous pocket formation that is based on binary collision simulations to generate the distribution of deposited energy, and on numerical solution of the heat transport equation to describe the quenching process. The heat transport equation is modified to consider the heat of melting when the melting temperature is crossed at any point in space. It is discretized with finite differences on grid points that coincide with the crystallographic lattice sites, which allows easy determination of molten atoms. Atoms are considered molten if the average of their energy and the energy of their neighbors meets the melting criterion. The results obtained with this model are in good overall agreement with published experimental data on P, As, Te and Tl implantations in Si and with data on the polyatomic effect at cryogenic temperature
Numerical multistep methods for the efficient solution of quantum mechanics and related problems
International Nuclear Information System (INIS)
Anastassi, Z.A.; Simos, T.E.
2009-01-01
In this paper we present the recent development in the numerical integration of the Schroedinger equation and related systems of ordinary differential equations with oscillatory solutions, such as the N-body problem. We examine several types of multistep methods (explicit, implicit, predictor-corrector, hybrid) and several properties (P-stability, trigonometric fitting of various orders, phase fitting, high phase-lag order, algebraic order). We analyze the local truncation error and the stability of the methods. The error for the Schroedinger equation is also presented, which reveals the relation of the error to the energy. The efficiency of the methods is evaluated through the integration of five problems. Figures are presented and analyzed and some general conclusions are made. Code written in Maple is given for the development of all methods analyzed in this paper. Also the subroutines written in Matlab, that concern the integration of the methods, are presented.
Numerical Solution of Nonlinear Volterra Integral Equations System Using Simpson’s 3/8 Rule
Directory of Open Access Journals (Sweden)
Adem Kılıçman
2012-01-01
Full Text Available The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of order O(h4. Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.
Penalty methods for the numerical solution of American multi-asset option problems
Nielsen, Bjørn Fredrik; Skavhaug, Ola; Tveito, Aslak
2008-12-01
We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.
Cremer, Clemens; Neuweiler, Insa; Bechtold, Michel; Vanderborght, Jan
2016-04-01
Quantification of flow and solute transport in the shallow subsurface adjacent to the atmosphere is decisive to prevent groundwater pollution and conserve groundwater quality, to develop successful remediation strategies and to understand nutrient cycling. In nature, due to erratic precipitation-evaporation patterns, soil moisture content and related hydraulic conductivity in the vadose zone are not only variable in space but also in time. Flow directions and flow paths locally change between precipitation and evaporation periods. This makes the identification and description of solute transport processes in the vadose zone a complex problem. Recent studies (Lehmann and Or, 2009; Bechtold et al., 2011a) focused on the investigation of upward transport of solutes during evaporation in heterogeneous soil columns, where heterogeneity was introduced by a sharp vertical material interface between two types of sand. Lateral solute transport through the interface in both (lateral) directions was observed at different depths of the investigated soil columns. Following recent approaches, we conduct two-dimensional numerical simulations in a similar setup which is composed of two sands with a sharp vertical material interface. The investigation is broadened from the sole evaporation to combined precipitation-evaporation cycles in order to quantify transport processes resulting from the combined effects of heterogeneous soil structure and dynamic flow conditions. Simulations are performed with a coupled finite volume and random walk particle tracking algorithm (Ippisch et al., 2006; Bechtold et al., 2011b). By comparing scenarios with cyclic boundary conditions and stationary counterparts with the same net flow rate, we found that duration and intensity of precipitation and evaporation periods potentially have an influence on lateral redistribution of solutes and thus leaching rates. Whether or not dynamic boundary conditions lead to significant deviations in the transport
International Nuclear Information System (INIS)
Matthes, W.K.
1998-01-01
The 'adjoint transport equation in its integro-differential form' is derived for the radiation damage produced by atoms injected into solids. We reduce it to the one-dimensional form and prepare it for a numerical solution by: --discretizing the continuous variables energy, space and direction, --replacing the partial differential quotients by finite differences and --evaluating the collision integral by a double sum. By a proper manipulation of this double sum the adjoint transport equation turns into a (very large) set of linear equations with tridiagonal matrix which can be solved by a special (simple and fast) algorithm. The solution of this set of linear equations contains complete information on a specified damage type (e.g. the energy deposited in a volume V) in terms of the function D(i,E,c,x) which gives the damage produced by all particles generated in a cascade initiated by a particle of type i starting at x with energy E in direction c. It is essential to remark that one calculation gives the damage function D for the complete ranges of the variables {i,E,c and x} (for numerical reasons of course on grid-points in the {E,c,x}-space). This is most useful to applications where a general source-distribution S(i,E,c,x) of particles is given by the experimental setup (e.g. beam-window and and target in proton accelerator work. The beam-protons along their path through the window--or target material generate recoil atoms by elastic collisions or nuclear reactions. These recoil atoms form the particle source S). The total damage produced then is eventually given by: D = (Σ)i ∫ ∫ ∫ S(i, E, c, x)*D(i, E, c, x)*dE*dc*dx A Fortran-77 program running on a PC-486 was written for the overall procedure and applied to some problems
Rienstra, S.W.; Eversman, W.
2001-01-01
An explicit, analytical, multiple-scales solution for modal sound transmission through slowly varying ducts with mean flow and acoustic lining is tested against a numerical finite-element solution solving the same potential flow equations. The test geometry taken is representative of a high-bypass
DEFF Research Database (Denmark)
Celia, Michael A.; Binning, Philip John
1992-01-01
that the algorithm produces solutions that are essentially mass conservative and oscillation free, even in the presence of steep infiltrating fronts. When the algorithm is applied to the case of air and water flow in unsaturated soils, numerical results confirm the conditions under which Richards's equation is valid....... Numerical results also demonstrate the potential importance of air phase advection when considering contaminant transport in unsaturated soils. Comparison to several other numerical algorithms shows that the modified Picard approach offers robust, mass conservative solutions to the general equations...
Features of the Numerical Solution of Thermal Destruction Fuel Pins Problems in the Fast Reactor
Usov, E. V.; Butov, A. A.; Klimonov, I. A.; Chuhno, V. I.; Nikolaenko, A. V.; Zhdanov, V. S.; Pribaturin, N. A.; Strizhov, V. F.
2017-11-01
In this paper the description of the basic equations which can be used for calculation of melting of fuel and cladding of the fast reactor, moving of the melt on a fuel pin surface and its solidification is presented. The special attention is given speed of calculation algorithms and fidelity of the phenomena which are observed at a stage of severe accidents in fast reactors. For check of working capacity of initial models, numerical calculations of Stefan-type problems on front movement of melting/solidification in cylindrical geometry are presented. Comparison with the solutions received by known analytical methods is executed. For validation of the numerical realization of calculation algorithms the analysis is carried out and experiments in which melting of the model fuel pins of fast reactors was studied are chosen. On the basis of the chosen experiments calculation schemes taking into account initial and boundary conditions are prepared and modeling is performed. Modeling results are shown in the present paper. Estimation of calculation error of the basic physical parameters is done by results of the modeling and conclusions are drawn on a correctness of algorithms operation.
Numerical path integral solution to strong Coulomb correlation in one dimensional Hooke's atom
Ruokosenmäki, Ilkka; Gholizade, Hossein; Kylänpää, Ilkka; Rantala, Tapio T.
2017-01-01
We present a new approach based on real time domain Feynman path integrals (RTPI) for electronic structure calculations and quantum dynamics, which includes correlations between particles exactly but within the numerical accuracy. We demonstrate that incoherent propagation by keeping the wave function real is a novel method for finding and simulation of the ground state, similar to Diffusion Monte Carlo (DMC) method, but introducing new useful tools lacking in DMC. We use 1D Hooke's atom, a two-electron system with very strong correlation, as our test case, which we solve with incoherent RTPI (iRTPI) and compare against DMC. This system provides an excellent test case due to exact solutions for some confinements and because in 1D the Coulomb singularity is stronger than in two or three dimensional space. The use of Monte Carlo grid is shown to be efficient for which we determine useful numerical parameters. Furthermore, we discuss another novel approach achieved by combining the strengths of iRTPI and DMC. We also show usefulness of the perturbation theory for analytical approximates in case of strong confinements.
Salpeter equation in position space: Numerical solution for arbitrary confining potentials
International Nuclear Information System (INIS)
Nickisch, L.J.; Durand, L.; Durand, B.
1984-01-01
We present and test two new methods for the numerical solution of the relativistic wave equation [(-del 2 +m 1 2 )/sup 1/2/+(-del 2 +m 2 2 )/sup 1/2/+V(r)-M]psi( r ) = 0, which appears in the theory of relativistic quark-antiquark bound states. Our methods work directly in position space, and hence have the desirable features that we can vary the potential V(r) locally in fitting the qq-bar mass spectrum, and can easily build in the expected behavior of V for r→0,infinity. Our first method converts the nonlocal square-root operators to mildly singular integral operators involving hyperbolic Bessel functions. The resulting integral equation can be solved numerically by matrix techniques. Our second method approximates the square-root operators directly by finite matrices. Both methods converge rapidly with increasing matrix size (the square-root matrix method more rapidly) and can be used in fast-fitting routines. We present some tests for oscillator and Coulomb interactions, and for the realistic Coulomb-plus-linear potential used in qq-bar phenomenology
International Nuclear Information System (INIS)
Yamamoto, Akio; Tatsumi, Masahiro; Sugimura, Naoki
2007-01-01
The Krylov subspace method is applied to solve nuclide burnup equations used for lattice physics calculations. The Krylov method is an efficient approach for solving ordinary differential equations with stiff nature such as the nuclide burnup with short lived nuclides. Some mathematical fundamentals of the Krylov subspace method and its application to burnup equations are discussed. Verification calculations are carried out in a PWR pin-cell geometry with UO 2 fuel. A detailed burnup chain that includes 193 fission products and 28 heavy nuclides is used in the verification calculations. Shortest half life found in the present burnup chain is approximately 30 s ( 106 Rh). Therefore, conventional methods (e.g., the Taylor series expansion with scaling and squaring) tend to require longer computation time due to numerical stiffness. Comparison with other numerical methods (e.g., the 4-th order Runge-Kutta-Gill) reveals that the Krylov subspace method can provide accurate solution for a detailed burnup chain used in the present study with short computation time. (author)
Directory of Open Access Journals (Sweden)
Aydin Secer
2013-01-01
Full Text Available An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions.
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Ravi Borana
2016-09-01
Full Text Available In the petroleum reservoir at an early stage the oil is recovered due to existing natural pressure and such type of oil recovery is referred as primary oil recovery. It ends when pressure equilibrium occurs and still large amount of oil remains in the reservoir. Consequently, secondary oil recovery process is employed by injection water into some injection wells to push oil towards the production well. The instability phenomenon arises during secondary oil recovery process. When water is injected into the oil filled region, due to the force of injecting water and difference in viscosities of water and native oil, protuberances occur at the common interface. It gives rise to the shape of fingers (protuberances at common interface. The injected water shoots through inter connected capillaries at very high speed. It appears in the form of irregular trembling fingers, filled with injected water in the native oil field; this is due to the immiscibility of water and oil. The homogeneous porous medium is considered with a small inclination with the horizontal, the basic parameters porosity and permeability remain uniform throughout the porous medium. Based on the mass conservation principle and important Darcy's law under the specific standard relationships and basic assumptions considered, the governing equation yields a non-linear partial differential equation. The Crank–Nicolson finite difference scheme is developed and on implementing the boundary conditions the resulting finite difference scheme is implemented to obtain the numerical results. The numerical results are obtained by generating a MATLAB code for the saturation of water which decreases with the space variable and increases with time. The obtained numerical solution is efficient, accurate, and reliable, matches well with the physical phenomenon.
Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em
2017-12-01
Starting with the asymptotic expansion of the error equation of the shifted Gr\\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions. We show theoretically and numerically that numerical solutions up to certain accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new formula leads to better performance compared to other known numerical approximations with similar resolution.
Two numerical methods for the solution of two-dimensional eddy current problems
International Nuclear Information System (INIS)
Biddlecombe, C.S.
1978-07-01
A general method for the solution of eddy current problems in two dimensions - one component of current density and two of magnetic field, is reported. After examining analytical methods two numerical methods are presented. Both solve the two dimensional, low frequency limit of Maxwell's equations for transient eddy currents in conducting material, which may be permeable, in the presence of other non-conducting permeable material. Both solutions are expressed in terms of the magnetic vector potential. The first is an integral equation method, using zero order elements in the discretisation of the unknown source regions. The other is a differential equation method, using a first order finite element mesh, and the Galerkin weighted residual procedure. The resulting equations are solved as initial-value problems. Results from programs based on each method are presented showing the power and limitations of the methods and the range of problems solvable. The methods are compared and recommendations are made for choosing between them. Suggestions are made for improving both methods, involving boundary integral techniques. (author)
New numerical methods for open-loop and feedback solutions to dynamic optimization problems
Ghosh, Pradipto
The topic of the first part of this research is trajectory optimization of dynamical systems via computational swarm intelligence. Particle swarm optimization is a nature-inspired heuristic search method that relies on a group of potential solutions to explore the fitness landscape. Conceptually, each particle in the swarm uses its own memory as well as the knowledge accumulated by the entire swarm to iteratively converge on an optimal or near-optimal solution. It is relatively straightforward to implement and unlike gradient-based solvers, does not require an initial guess or continuity in the problem definition. Although particle swarm optimization has been successfully employed in solving static optimization problems, its application in dynamic optimization, as posed in optimal control theory, is still relatively new. In the first half of this thesis particle swarm optimization is used to generate near-optimal solutions to several nontrivial trajectory optimization problems including thrust programming for minimum fuel, multi-burn spacecraft orbit transfer, and computing minimum-time rest-to-rest trajectories for a robotic manipulator. A distinct feature of the particle swarm optimization implementation in this work is the runtime selection of the optimal solution structure. Optimal trajectories are generated by solving instances of constrained nonlinear mixed-integer programming problems with the swarming technique. For each solved optimal programming problem, the particle swarm optimization result is compared with a nearly exact solution found via a direct method using nonlinear programming. Numerical experiments indicate that swarm search can locate solutions to very great accuracy. The second half of this research develops a new extremal-field approach for synthesizing nearly optimal feedback controllers for optimal control and two-player pursuit-evasion games described by general nonlinear differential equations. A notable revelation from this development
Hasani, Mojtaba H; Gharibzadeh, Shahriar; Farjami, Yaghoub; Tavakkoli, Jahan
2013-09-01
Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.
Directory of Open Access Journals (Sweden)
Suheel Abdullah Malik
2014-01-01
Full Text Available We present a hybrid heuristic computing method for the numerical solution of nonlinear singular boundary value problems arising in physiology. The approximate solution is deduced as a linear combination of some log sigmoid basis functions. A fitness function representing the sum of the mean square error of the given nonlinear ordinary differential equation (ODE and its boundary conditions is formulated. The optimization of the unknown adjustable parameters contained in the fitness function is performed by the hybrid heuristic computation algorithm based on genetic algorithm (GA, interior point algorithm (IPA, and active set algorithm (ASA. The efficiency and the viability of the proposed method are confirmed by solving three examples from physiology. The obtained approximate solutions are found in excellent agreement with the exact solutions as well as some conventional numerical solutions.
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Qicheng Meng
2016-04-01
Full Text Available A third-order KdV solution to the internal solitary wave is derived by a new method based on the weakly nonlinear assumptions in a rigid-lid two-layer system. The solution corrects an error by Mirie and Su (1984. A two-dimensional numerical wave tank has been established with the help of the open source CFD library OpenFOAM and the third-party software waves2Foam. Various analytical solutions, including the first-order to third-order KdV solutions, the eKdV solution and the MCC solution, have been used to initialise the flow fields in the CFD simulations of internal solitary waves. Two groups including 11 numerical cases have been carried out. In the same group, the initial wave amplitudes are the same but the implemented analytical solutions are different. The simulated wave profiles at different moments have been presented. The relative errors in terms of the wave amplitude between the last time step and the initial input have been analysed quantitatively. It is found that the third-order KdV solution results in the most stable internal solitary wave in the numerical wave tank for both small-amplitude and finite-amplitude cases. The finding is significant for the further simulations involving internal solitary waves.
D'Ambrosio, Raffaele; Moccaldi, Martina; Paternoster, Beatrice
2018-05-01
In this paper, an adapted numerical scheme for reaction-diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit-explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.
International Nuclear Information System (INIS)
Kupka, F.
1997-11-01
This thesis deals with the extension of sparse grid techniques to spectral methods for the solution of partial differential equations with periodic boundary conditions. A review on boundary and initial-boundary value problems and a discussion on numerical resolution is used to motivate this research. Spectral methods are introduced by projection techniques, and by three model problems: the stationary and the transient Helmholtz equations, and the linear advection equation. The approximation theory on the hyperbolic cross is reviewed and its close relation to sparse grids is demonstrated. This approach extends to non-periodic problems. Various Sobolev spaces with dominant mixed derivative are introduced to provide error estimates for Fourier approximation and interpolation on the hyperbolic cross and on sparse grids by means of Sobolev norms. The theorems are immediately applicable to the stability and convergence analysis of sparse grid spectral methods. This is explicitly demonstrated for the three model problems. A variant of the von Neumann condition is introduced to simplify the stability analysis of the time-dependent model problems. The discrete Fourier transformation on sparse grids is discussed together with its software implementation. Results on numerical experiments are used to illustrate the performance of the new method with respect to the smoothness properties of each example. The potential of the method in mathematical modelling is estimated and generalizations to other sparse grid methods are suggested. The appendix includes a complete Fortran90 program to solve the linear advection equation by the sparse grid Fourier collocation method and a third-order Runge-Kutta routine for integration in time. (author)
Numerical Upscaling of Solute Transport in Fractured Porous Media Based on Flow Aligned Blocks
Leube, P.; Nowak, W.; Sanchez-Vila, X.
2013-12-01
High-contrast or fractured-porous media (FPM) pose one of the largest unresolved challenges for simulating large hydrogeological systems. The high contrast in advective transport between fast conduits and low-permeability rock matrix, including complex mass transfer processes, leads to the typical complex characteristics of early bulk arrivals and long tailings. Adequate direct representation of FPM requires enormous numerical resolutions. For large scales, e.g. the catchment scale, and when allowing for uncertainty in the fracture network architecture or in matrix properties, computational costs quickly reach an intractable level. In such cases, multi-scale simulation techniques have become useful tools. They allow decreasing the complexity of models by aggregating and transferring their parameters to coarser scales and so drastically reduce the computational costs. However, these advantages come at a loss of detail and accuracy. In this work, we develop and test a new multi-scale or upscaled modeling approach based on block upscaling. The novelty is that individual blocks are defined by and aligned with the local flow coordinates. We choose a multi-rate mass transfer (MRMT) model to represent the remaining sub-block non-Fickian behavior within these blocks on the coarse scale. To make the scale transition simple and to save computational costs, we capture sub-block features by temporal moments (TM) of block-wise particle arrival times to be matched with the MRMT model. By predicting spatial mass distributions of injected tracers in a synthetic test scenario, our coarse-scale solution matches reasonably well with the corresponding fine-scale reference solution. For predicting higher TM-orders (such as arrival time and effective dispersion), the prediction accuracy steadily decreases. This is compensated to some extent by the MRMT model. If the MRMT model becomes too complex, it loses its effect. We also found that prediction accuracy is sensitive to the choice of
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Gisele Tessari Santos
2009-08-01
Full Text Available A large number of financial engineering problems involve non-linear equations with non-linear or time-dependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.Muitos problemas de engenharia financeira envolvem equações não-lineares com condições de contorno não-lineares ou dependentes do tempo. Apesar de soluções analíticas disponíveis, várias formas clássicas e modificadas da conhecida equação de Black-Scholes (BS requerem soluções numéricas rápidas e acuradas. Este trabalho introduz o método de função de base radial (RBF aplicado à solução da equação BS com condições de contorno não-lineares relacionadas a opções de barreira dependentes da trajetória. Além disso, explora-se o método difusional para solucionar equações advectivo-difusivas quanto à sua efetividade para solucionar equações BS. Utilizam-se funções de base radial Cúbica e Thin-Plate Spline (TPS, aplicadas à solução de problemas de opções de barreiras. Os resultados numéricos, quando comparados com as soluções analíticas, permitem afirmar
International Nuclear Information System (INIS)
Vasileva, D.P.
1993-01-01
Blow-up and global time self-similar solutions of a boundary problem for a nonlinear equation u t = Δ u σ+1 + u β are found in the case β = σ + 1. It is shown that they describe the asymptotic behavior of a wide class of initial perturbations. A numerical investigation of the solutions in the case β>σ + 1 is also made. A hypothesis is done that the behavior for large times of global time solutions is described by the self-similar solutions of the equation without source.(author). 20 refs.; 9 figs
On the formation, growth, and shapes of solution pipes - insights from numerical modeling
Szymczak, Piotr; Tredak, Hanna; Upadhyay, Virat; Kondratiuk, Paweł; Ladd, Anthony J. C.
2015-04-01
Cylindrical, vertical structures called solution pipes are a characteristic feature of epikarst, encountered in different parts of the world, both in relatively cold areas such as England and Poland (where their formation is linked to glacial processes) [1] and in coastal areas in tropical or subtropical climate (Bermuda, Australia, South Africa, Caribbean, Mediterranean) [2,3]. They are invariably associated with weakly cemented, porous limestones and relatively high groundwater fluxes. Many of them develop under the colluvial sandy cover and contain the fill of clayey silt. Although it is widely accepted that they are solutional in origin, the exact mechanism by which the flow becomes focused is still under debate. The hypotheses include the concentration of acidified water around stems and roots of plants, or the presence of pre-existing fractures or steeply dipping bedding planes, which would determine the points of entry for the focused groundwater flows. However, there are field sites where neither of this mechanisms was apparently at play and yet the pipes are formed in large quantities [1]. In this communication we show that the systems of solution pipes can develop spontaneously in nearly uniform matrix due to the reactive-infiltration instability: a homogeneous porous matrix is unstable with respect to small variations in local permeability; regions of high permeability dissolve faster because of enhanced transport of reactants, which leads to increased rippling of the front. This leads to the formation of a system of solution pipes which then advance into the matrix. We study this process numerically, by a combination of 2d- and 3d-simulations, solving the coupled flow and transport equations at the Darcy scale. The relative simplicity of this system (pipes developing in a uniform porous matrix, without any pre-existing structure) makes it very attractive from the modeling standpoint. We quantify the factors which control the pipe diameters and the
Russo, David; Laufer, Asher; Shapira, Roi H.; Kurtzman, Daniel
2013-02-01
Detailed numerical simulations were used to analyze water flow and transport of nitrate, chloride, and a tracer solute in a 3-D, spatially heterogeneous, variably saturated soil, originating from a citrus orchard irrigated with treated sewage water (TSW) considering realistic features of the soil-water-plant-atmosphere system. Results of this study suggest that under long-term irrigation with TSW, because of nitrate uptake by the tree roots and nitrogen transformations, the vadose zone may provide more capacity for the attenuation of the nitrate load in the groundwater than for the chloride load in the groundwater. Results of the 3-D simulations were used to assess their counterparts based on a simplified, deterministic, 1-D vertical simulation and on limited soil monitoring. Results of the analyses suggest that the information that may be gained from a single sampling point (located close to the area active in water uptake by the tree roots) or from the results of the 1-D simulation is insufficient for a quantitative description of the response of the complicated, 3-D flow system. Both might considerably underestimate the movement and spreading of a pulse of a tracer solute and also the groundwater contamination hazard posed by nitrate and particularly by chloride moving through the vadose zone. This stems mainly from the rain that drove water through the flow system away from the rooted area and could not be represented by the 1-D model or by the single sampling point. It was shown, however, that an additional sampling point, located outside the area active in water uptake, may substantially improve the quantitative description of the response of the complicated, 3-D flow system.
Yao, Lingxing; Mori, Yoichiro
2017-12-01
Osmotic forces and solute diffusion are increasingly seen as playing a fundamental role in cell movement. Here, we present a numerical method that allows for studying the interplay between diffusive, osmotic and mechanical effects. An osmotically active solute obeys a advection-diffusion equation in a region demarcated by a deformable membrane. The interfacial membrane allows transmembrane water flow which is determined by osmotic and mechanical pressure differences across the membrane. The numerical method is based on an immersed boundary method for fluid-structure interaction and a Cartesian grid embedded boundary method for the solute. We demonstrate our numerical algorithm with the test case of an osmotic engine, a recently proposed mechanism for cell propulsion.
Lee, Jonghyun; Rolle, Massimo; Kitanidis, Peter K
2017-09-15
Most recent research on hydrodynamic dispersion in porous media has focused on whole-domain dispersion while other research is largely on laboratory-scale dispersion. This work focuses on the contribution of a single block in a numerical model to dispersion. Variability of fluid velocity and concentration within a block is not resolved and the combined spreading effect is approximated using resolved quantities and macroscopic parameters. This applies whether the formation is modeled as homogeneous or discretized into homogeneous blocks but the emphasis here being on the latter. The process of dispersion is typically described through the Fickian model, i.e., the dispersive flux is proportional to the gradient of the resolved concentration, commonly with the Scheidegger parameterization, which is a particular way to compute the dispersion coefficients utilizing dispersivity coefficients. Although such parameterization is by far the most commonly used in solute transport applications, its validity has been questioned. Here, our goal is to investigate the effects of heterogeneity and mass transfer limitations on block-scale longitudinal dispersion and to evaluate under which conditions the Scheidegger parameterization is valid. We compute the relaxation time or memory of the system; changes in time with periods larger than the relaxation time are gradually leading to a condition of local equilibrium under which dispersion is Fickian. The method we use requires the solution of a steady-state advection-dispersion equation, and thus is computationally efficient, and applicable to any heterogeneous hydraulic conductivity K field without requiring statistical or structural assumptions. The method was validated by comparing with other approaches such as the moment analysis and the first order perturbation method. We investigate the impact of heterogeneity, both in degree and structure, on the longitudinal dispersion coefficient and then discuss the role of local dispersion
A numerical solution of the problem of crown forest fire initiation and spread
Marzaeva, S. I.; Galtseva, O. V.
2018-05-01
Mathematical model of forest fire was based on an analysis of known experimental data and using concept and methods from reactive media mechanics. The study takes in to account the mutual interaction of the forest fires and three-dimensional atmosphere flows. The research is done by means of mathematical modeling of physical processes. It is based on numerical solution of Reynolds equations for chemical components and equations of energy conservation for gaseous and condensed phases. It is assumed that the forest during a forest fire can be modeled as a two-temperature multiphase non-deformable porous reactive medium. A discrete analog for the system of equations was obtained by means of the control volume method. The developed model of forest fire initiation and spreading would make it possible to obtain a detailed picture of the variation in the velocity, temperature and chemical species concentration fields with time. Mathematical model and the result of the calculation give an opportunity to evaluate critical conditions of the forest fire initiation and spread which allows applying the given model for of means for preventing fires.
Low Mach number analysis of idealized thermoacoustic engines with numerical solution.
Hireche, Omar; Weisman, Catherine; Baltean-Carlès, Diana; Le Quéré, Patrick; Bauwens, Luc
2010-12-01
A model of an idealized thermoacoustic engine is formulated, coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load. Correct coupling results in an asymptotically consistent global model, in the small Mach number approximation. A well-resolved numerical solution is obtained for two-dimensional heat exchangers and stack. The model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative Mach number. The model is well-suited for simulation of the entire startup process, whereby as a result of some excitation, an initially specified temperature profile in the stack evolves toward a near-steady profile, eventually reaching stationary operation. A validation analysis is presented, together with results showing the early amplitude growth and approach of a stationary regime. Two types of initial excitation are used: Random noise and a small periodic wave. The set of assumptions made leads to a heat-exchanger section that acts as a source of volume but is transparent to pressure and to a local heat-exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed.
A conservative finite difference method for the numerical solution of plasma fluid equations
International Nuclear Information System (INIS)
Colella, P.; Dorr, M.R.; Wake, D.D.
1999-01-01
This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level
Amitai, Dganit; Averbuch, Amir; Itzikowitz, Samuel; Turkel, Eli
1991-01-01
A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent process. Removing the synchronization constraint has the potential of speeding up the computation. The authors present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although the discussion concentrates on the Euler scheme for the solution of the heat equation, it has the potential for being extended to other schemes and other parabolic partial differential equations (PDEs). These schemes are analyzed and implemented on the shared memory multi-user Sequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that the synchronization penalty can be about 50 percent of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time level, and with the problem dimension. The efficiency of the AS may reach 90 percent and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient, but provides more accurate results for intermediate (non steady-state) values.
Numerical solutions of the aerosol general dynamic equation for nuclear reactor safety studies
International Nuclear Information System (INIS)
Park, J.W.
1988-01-01
Methods and approximations inherent in modeling of aerosol dynamics and evolution for nuclear reactor source term estimation have been investigated. Several aerosol evolution problems are considered to assess numerical methods of solving the aerosol dynamic equation. A new condensational growth model is constructed by generalizing Mason's formula to arbitrary particle sizes, and arbitrary accommodation of the condensing vapor and background gas at particle surface. Analytical solution is developed for the aerosol growth equation employing the new condensation model. The space-dependent aerosol dynamic equation is solved to assess implications of spatial homogenization of aerosol distributions. The results of our findings are as follows. The sectional method solving the aerosol dynamic equation is quite efficient in modeling of coagulation problems, but should be improved for simulation of strong condensation problems. The J-space transform method is accurate in modeling of condensation problems, but is very slow. For the situation considered, the new condensation model predicts slower aerosol growth than the corresponding isothermal model as well as Mason's model, the effect of partial accommodation is considerable on the particle evolution, and the effect of the energy accommodation coefficient is more pronounced than that of the mass accommodation coefficient. For the initial conditions considered, the space-dependent aerosol dynamics leads to results that are substantially different from those based on the spatially homogeneous aerosol dynamic equation
Evaluate the accuracy of the numerical solution of hydrogeological problems of mass transfer
Directory of Open Access Journals (Sweden)
Yevhrashkina G.P.
2014-12-01
Full Text Available In the hydrogeological task on quantifying pollution of aquifers the error are starting add up with moment organization of regime observation network as a source of information on the pollution of groundwater in order to evaluate migration options for future prognosis calculations. Optimum element regime observation network should consist of three drill holes on the groundwater flow at equal distances from one another and transversely to the flow of the three drill holes, and at equal distances. If the target of observation drill holes coincides with the stream line on which will then be decided by direct migration task, the error will be minimal. The theoretical basis and results of numerical experiments to assess the accuracy of direct predictive tasks planned migration of groundwater in the area of full water saturation. For the vadose zone, we consider problems of vertical salt transport moisture. All studies were performed by comparing the results of fundamental and approximate solutions in a wide range of characteristics of the processes, which are discussed in relation to ecological and hydrogeological conditions of mining regions on the example of the Western Donbass.
Reduced-Order Direct Numerical Simulation of Solute Transport in Porous Media
Mehmani, Yashar; Tchelepi, Hamdi
2017-11-01
Pore-scale models are an important tool for analyzing fluid dynamics in porous materials (e.g., rocks, soils, fuel cells). Current direct numerical simulation (DNS) techniques, while very accurate, are computationally prohibitive for sample sizes that are statistically representative of the porous structure. Reduced-order approaches such as pore-network models (PNM) aim to approximate the pore-space geometry and physics to remedy this problem. Predictions from current techniques, however, have not always been successful. This work focuses on single-phase transport of a passive solute under advection-dominated regimes and delineates the minimum set of approximations that consistently produce accurate PNM predictions. Novel network extraction (discretization) and particle simulation techniques are developed and compared to high-fidelity DNS simulations for a wide range of micromodel heterogeneities and a single sphere pack. Moreover, common modeling assumptions in the literature are analyzed and shown that they can lead to first-order errors under advection-dominated regimes. This work has implications for optimizing material design and operations in manufactured (electrodes) and natural (rocks) porous media pertaining to energy systems. This work was supported by the Stanford University Petroleum Research Institute for Reservoir Simulation (SUPRI-B).
International Nuclear Information System (INIS)
Delfin L, A.
1996-01-01
The purpose of this work is to solve the neutron transport equation in discrete-ordinates and X-Y geometry by developing and using the strong discontinuous and strong modified discontinuous nodal finite element schemes. The strong discontinuous and modified strong discontinuous nodal finite element schemes go from two to ten interpolation parameters per cell. They are describing giving a set D c and polynomial space S c corresponding for each scheme BDMO, RTO, BL, BDM1, HdV, BDFM1, RT1, BQ and BDM2. The solution is obtained solving the neutron transport equation moments for each nodal scheme by developing the basis functions defined by Pascal triangle and the Legendre moments giving in the polynomial space S c and, finally, looking for the non singularity of the resulting linear system. The linear system is numerically solved using a computer program for each scheme mentioned . It uses the LU method and forward and backward substitution and makes a partition of the domain in cells. The source terms and angular flux are calculated, using the directions and weights associated to the S N approximation and solving the angular flux moments to find the effective multiplication constant. The programs are written in Fortran language, using the dynamic allocation of memory to increase efficiently the available memory of the computing equipment. (Author)
DEFF Research Database (Denmark)
including convection-difmsion-reaction PDEs are numerically solved using the two methods on the same spatial grid. Even though the CE/SE method uses a simple stencil structure and is developed on a simple mathematical basis (i.e., Gauss' divergence theorem), accurate and computationally-efficient solutions...
Analytical-numerical solution of a nonlinear integrodifferential equation in econometrics
Kakhktsyan, V. M.; Khachatryan, A. Kh.
2013-07-01
A mixed problem for a nonlinear integrodifferential equation arising in econometrics is considered. An analytical-numerical method is proposed for solving the problem. Some numerical results are presented.
Numerical solution of integral equations, describing mass spectrum of vector mesons
International Nuclear Information System (INIS)
Zhidkov, E.P.; Nikonov, E.G.; Sidorov, A.V.; Skachkov, N.B.; Khoromskij, B.N.
1988-01-01
The description of the numerical algorithm for solving quasipotential integral equation in impulse space is presented. The results of numerical computations of the vector meson mass spectrum and the leptonic decay width are given in comparison with the experimental data
International Nuclear Information System (INIS)
Anastassi, Z. A.; Simos, T. E.
2010-01-01
We develop a new family of explicit symmetric linear multistep methods for the efficient numerical solution of the Schroedinger equation and related problems with oscillatory solution. The new methods are trigonometrically fitted and have improved intervals of periodicity as compared to the corresponding classical method with constant coefficients and other methods from the literature. We also apply the methods along with other known methods to real periodic problems, in order to measure their efficiency.
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes
International Nuclear Information System (INIS)
Larsen, E.W.; Morel, J.E.; Miller, W.F. Jr.
1987-01-01
We present an asymptotic analysis of spatial differencing schemes for the discrete-ordinates equations, for diffusive media with spatial cells that are not optically thin. Our theoretical tool is an asymptotic expansion that has previously been used to describe the transform from analytic transport to analytic diffusion theory for such media. To introduce this expansion and its physical rationale, we first describe it for the analytic discrete-ordinates equations. Then, we apply the expansion to the spatially discretized discrete-ordinates equations, with the spatial mesh scaled in either of two physically relevant ways such that the optical thickness of the spatial cells is not small. If the result of either expansion is a legitimate diffusion description for either the cell-averaged or cell-edge fluxes, then we say that the approximate flux has the appropriate diffusion limit; otherwise, we say it does not. We consider several transport differencing schemes that are applicable in neutron transport and thermal radiation applications. We also include numerical results which demonstrate the validity of our theory and show that differencing schemes that do have a particular diffusion limit are substantially more accurate, in the regime described by the limit, than those that do not. copyright 1987 Academic Press, Inc
Energy Technology Data Exchange (ETDEWEB)
Brown, L.F.; Ebinger, M.H.
1996-08-01
Four simple precipitation problems are solved to examine the use of numerical equilibrium codes. The study emphasizes concentrated solutions, assumes both ideal and nonideal solutions, and employs different databases and different activity-coefficient relationships. The study uses the EQ3/6 numerical speciation codes. The results show satisfactory material balances and agreement between solubility products calculated from free-energy relationships and those calculated from concentrations and activity coefficients. Precipitates show slightly higher solubilities when the solutions are regarded as nonideal than when considered ideal, agreeing with theory. When a substance may precipitate from a solution dilute in the precipitating substance, a code may or may not predict precipitation, depending on the database or activity-coefficient relationship used. In a problem involving a two-component precipitation, there are only small differences in the precipitate mass and composition between the ideal and nonideal solution calculations. Analysis of this result indicates that this may be a frequent occurrence. An analytical approach is derived for judging whether this phenomenon will occur in any real or postulated precipitation situation. The discussion looks at applications of this approach. In the solutes remaining after the precipitations, there seems to be little consistency in the calculated concentrations and activity coefficients. They do not appear to depend in any coherent manner on the database or activity-coefficient relationship used. These results reinforce warnings in the literature about perfunctory or mechanical use of numerical speciation codes.
Method of independent timesteps in the numerical solution of initial value problems
International Nuclear Information System (INIS)
Porter, A.P.
1976-01-01
In the numerical solution of initial-value problems in several independent variables, the timestep is controlled, especially in the presence of shocks, by a small portion of the logical mesh, what one may call the crisis zone. One is frustrated by the necessity of doing in the whole mesh frequent calculations required by only a small part of the mesh. It is shown that it is possible to choose different timesteps natural to different parts of the mesh and to advance each zone in time only as often as is appropriate to that zone's own natural timestep. Prior work is reviewed and for the first time an investigation of the conditions for well-posedness, consistency and stability in independent timesteps is presented; a new method results. The prochronic and parachronic Cauchy surfaces are identified; and the reasons (well-posedness) for constraining the Cauchy surfaces to be prochronic (as distinct from the method of Grandey), that is, to lie prior to the time of the crisis zone (the zone of least timestep), are indicated. Stability (in the maximum norm) of parabolic equations and (in the L2 norm) of hyperbolic equations is reviewed, without restricting the treatment to linear equations or constant coefficients, and stability of the new method is proven in this framework. The details of the method of independent timesteps, the rules for choosing timesteps and for deciding when to update and when to skip zones, and the method of joining adjacent regions of differing timestep are described. The stability of independent timestep difference schemes is analyzed and exhibited. The economic advantages of the method, which often amount to an order-of-magnitude decrease in running time relative to conventional or implicit difference methods, are noted
Method of independent timesteps in the numerical solution of initial value problems
Energy Technology Data Exchange (ETDEWEB)
Porter, A.P.
1976-07-21
In the numerical solution of initial-value problems in several independent variables, the timestep is controlled, especially in the presence of shocks, by a small portion of the logical mesh, what one may call the crisis zone. One is frustrated by the necessity of doing in the whole mesh frequent calculations required by only a small part of the mesh. It is shown that it is possible to choose different timesteps natural to different parts of the mesh and to advance each zone in time only as often as is appropriate to that zone's own natural timestep. Prior work is reviewed and for the first time an investigation of the conditions for well-posedness, consistency and stability in independent timesteps is presented; a new method results. The prochronic and parachronic Cauchy surfaces are identified; and the reasons (well-posedness) for constraining the Cauchy surfaces to be prochronic (as distinct from the method of Grandey), that is, to lie prior to the time of the crisis zone (the zone of least timestep), are indicated. Stability (in the maximum norm) of parabolic equations and (in the L2 norm) of hyperbolic equations is reviewed, without restricting the treatment to linear equations or constant coefficients, and stability of the new method is proven in this framework. The details of the method of independent timesteps, the rules for choosing timesteps and for deciding when to update and when to skip zones, and the method of joining adjacent regions of differing timestep are described. The stability of independent timestep difference schemes is analyzed and exhibited. The economic advantages of the method, which often amount to an order-of-magnitude decrease in running time relative to conventional or implicit difference methods, are noted.
International Nuclear Information System (INIS)
Biyikoglu, A.; Akcayol, M.A.; Oezdemir, V.; Sivrioglu, M.
2005-01-01
In this study, steady state combustion in boilers with a fixed bed has been investigated. Temperature distributions in the combustion chamber of a coal fired boiler with a fixed bed are predicted using fuzzy logic based on data obtained from the numerical solution method for various coal and air feeding rates. The numerical solution method and the discretization of the governing equations of two dimensional turbulent flow in the combustion chamber and one dimensional coal combustion in the fixed bed are explained. Control Volume and Finite Difference Methods are used in the discretization of the equations in the combustion chamber and in the fixed bed, respectively. Results are presented as contours within the solution domain and compared with numerical ones. Comparison of the results shows that the difference between the numerical solution and fuzzy logic prediction throughout the computational domain is less than 1.5%. The statistical coefficient of multiple determinations for the investigated cases is about 0.9993 to 0.9998. This accuracy degree is acceptable in predicting the temperature values. So, it can be concluded that fuzzy logic provides a feasible method for defining the system properties
Botta, E.F.F.; Dijkstra, D.; Veldman, A.E.P.
1972-01-01
The numerical method of solution for the semi-infinite flat plate has been extended to the case of the parabolic cylinder. Results are presented for the skin friction, the friction drag, the pressure and the pressure drag. The drag coefficients have been checked by means of an application of the
Veldman, A.E.P.
1973-01-01
A numerical method is presented for the solution of the Navier-Stokes equations for flow past a paraboloid of revolution. The flow field has been computed for a large range of Reynolds numbers. Results are presented for the skinfriction and the pressure together with their respective drag
Felici, Helene M.; Drela, Mark
1993-01-01
A new approach based on the coupling of an Eulerian and a Lagrangian solver, aimed at reducing the numerical diffusion errors of standard Eulerian time-marching finite-volume solvers, is presented. The approach is applied to the computation of the secondary flow in two bent pipes and the flow around a 3D wing. Using convective point markers the Lagrangian approach provides a correction of the basic Eulerian solution. The Eulerian flow in turn integrates in time the Lagrangian state-vector. A comparison of coarse and fine grid Eulerian solutions makes it possible to identify numerical diffusion. It is shown that the Eulerian/Lagrangian approach is an effective method for reducing numerical diffusion errors.
Wu, Yang; Kelly, Damien P.
2014-12-01
The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of ? and ? type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of ? and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by ?, where ? is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.
Wu, Yang; Kelly, Damien P
2014-12-12
The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of [Formula: see text] and [Formula: see text] type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of [Formula: see text] and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by [Formula: see text], where [Formula: see text] is the replica order. These circular replicas are shown to be fundamentally
International Nuclear Information System (INIS)
Karabulut, Halit; Ipci, Duygu; Cinar, Can
2016-01-01
Highlights: • A numerical method has been developed for fully developed flows with constant wall temperature. • The governing equations were transformed to boundary fitted coordinates. • The Nusselt number of parabolic duct has been investigated. • Validation of the numerical method has been made by comparing published data. - Abstract: In motor-vehicles the use of more compact radiators have several advantages such as; improving the aerodynamic form of cars, reducing the weight and volume of the cars, reducing the material consumption and environmental pollutions, and enabling faster increase of the engine coolant temperature after starting to run and thereby improving the thermal efficiency. For the design of efficient and compact radiators, the robust determination of the heat transfer coefficient becomes imperative. In this study the external heat transfer coefficient of the radiator has been investigated for hydrodynamically and thermally fully developed flows in channels with constant wall temperature. In such situation the numerical treatment of the problem results in a trivial solution. To find a non-trivial solution the problem is treated either as an eigenvalue problem or as a thermally developing flow problem. In this study a numerical solution procedure has been developed and the heat transfer coefficients of the fully developed flow in triangular and parabolic air channels were investigated. The governing equations were transformed to boundary fitted coordinates and numerically solved. The non-trivial solution was obtained by means of guessing the temperature of any grid point within the solution domain. The correction of the guessed temperature was performed via smoothing the temperature profile on a line passing through the mentioned grid point. Results were compared with literature data and found to be consistent.
Directory of Open Access Journals (Sweden)
Panou G.
2017-02-01
Full Text Available The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.
Numerical solution of newton´s cooling differential equation by the methods of euler and runge-kutta
Directory of Open Access Journals (Sweden)
Andresa Pescador
2016-04-01
Full Text Available This article presents the first-order differential equations, which are a very important branch of mathematics as they have a wide applicability, in mathematics, as in physics, biology and economy. The objective of this study was to analyze the resolution of the equation that defines the cooling Newton's law. Verify its behavior using some applications that can be used in the classroom as an auxiliary instrument to the teacher in addressing these contents bringing answers to the questions of the students and motivating them to build their knowledge. It attempted to its resolution through two numerical methods, Euler method and Runge -Kutta method. Finally, there was a comparison of the approach of the solution given by the numerical solution with the analytical resolution whose solution is accurate.
Energy Technology Data Exchange (ETDEWEB)
Cobb, J.W.
1995-02-01
There is an increasing need for more accurate numerical methods for large-scale nonlinear magneto-fluid turbulence calculations. These methods should not only increase the current state of the art in terms of accuracy, but should also continue to optimize other desired properties such as simplicity, minimized computation, minimized memory requirements, and robust stability. This includes the ability to stably solve stiff problems with long time-steps. This work discusses a general methodology for deriving higher-order numerical methods. It also discusses how the selection of various choices can affect the desired properties. The explicit discussion focuses on third-order Runge-Kutta methods, including general solutions and five examples. The study investigates the linear numerical analysis of these methods, including their accuracy, general stability, and stiff stability. Additional appendices discuss linear multistep methods, discuss directions for further work, and exhibit numerical analysis results for some other commonly used lower-order methods.
Numerical Solution of Mixed Problems of the Theory of Elasticity with One-Sided Constraints
Directory of Open Access Journals (Sweden)
I. V. Stankevich
2017-01-01
enable us to ensure continuous approximation of not only displacements, but also stresses and strains. Mixed schemes to solve the boundary value problems lead to the saddle-point problems. Their solutions use various iterative techniques. One of the most effective techniques is a modified SSOR (MSSOR method, based on the SOR (Successive Over Relaxation one.The paper considers one of the options of the finite element method in the framework of mixed scheme that uses a Reissner functional. The procedures of the algorithm proposed in the paper are used to solve the problem of contact interaction when an elastic body of the finite dimensions, being under a load of the external forces, relies on the absolutely rigid half-space. The contact occurs with the distinguished contact surface, which in the general case can change its size during thermo-mechanical loading. The algorithm is implemented as an application software complex. The numerical study of the one-sided contact interaction between the elastic plate and the perfectly rigid half-space has shown a fairly high efficiency of the developed algorithm and the code that implements it.
Numerical Solution of Fuzzy Differential Equations by Runge-Kutta Verner Method
Directory of Open Access Journals (Sweden)
T. Jayakumar
2015-01-01
Full Text Available In this paper we study the numerical methods for Fuzzy Differential equations by an application of the Runge-Kutta Verner method for fuzzy differential equations. We prove a convergence result and give numerical examples to illustrate the theory.
Numerical Solution of the Blasius Viscous Flow Problem by Quartic B-Spline Method
Directory of Open Access Journals (Sweden)
Hossein Aminikhah
2016-01-01
Full Text Available A numerical method is proposed to study the laminar boundary layer about a flat plate in a uniform stream of fluid. The presented method is based on the quartic B-spline approximations with minimizing the error L2-norm. Theoretical considerations are discussed. The computed results are compared with some numerical results to show the efficiency of the proposed approach.
Khan, Sabeel M.; Sunny, D. A.; Aqeel, M.
2017-09-01
Nonlinear dynamical systems and their solutions are very sensitive to initial conditions and therefore need to be approximated carefully. In this article, we present and analyze nonlinear solution characteristics of the periodically forced Chen system with the application of a variational method based on the concept of finite time-elements. Our approach is based on the discretization of physical time space into finite elements where each time-element is mapped to a natural time space. The solution of the system is then determined in natural time space using a set of suitable basis functions. The numerical algorithm is presented and implemented to compute and analyze nonlinear behavior at different time-step sizes. The obtained results show an excellent agreement with the classical RK-4 and RK-5 methods. The accuracy and convergence of the method is shown by comparing numerically computed results with the exact solution for a test problem. The presented method has shown a great potential in dealing with the solutions of nonlinear dynamical systems and thus can be utilized in delineating different features and characteristics of their solutions.
Directory of Open Access Journals (Sweden)
Dmitry V. Lukyanenko
2017-01-01
Full Text Available This work develops a theory of the asymptotic-numerical investigation of the moving fronts in reaction-diffusion-advection models. By considering the numerical solution of the singularly perturbed Burgers’s equation we discuss a method of dynamically adapted mesh construction that is able to significantly improve the numerical solution of this type of equations. For the construction we use a priori information that is based on the asymptotic analysis of the problem. In particular, we take into account the information about the speed of the transition layer, its width and structure. Our algorithms are able to reduce significantly complexity and enhance stability of the numerical calculations in comparison with classical approaches for solving this class of problems. The numerical experiment is presented to demonstrate the effectiveness of the proposed method.The article is published in the authors’ wording.
Directory of Open Access Journals (Sweden)
Z. Pashazadeh Atabakan
2013-01-01
Full Text Available Spectral homotopy analysis method (SHAM as a modification of homotopy analysis method (HAM is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.
Energy Technology Data Exchange (ETDEWEB)
Teixeira, Vinicius Ligiero; Pires, Adolfo Puime; Bedrikovetsky, Pavel G. [Universidade Estadual do Norte Fluminense (UENF), Macae, RJ (Brazil). Lab. de Engenharia e Exploracao do Petroleo (LENEP)
2004-07-01
Enhanced Oil Recovery (EOR) methods include injection of different fluids into reservoirs to improve oil displacement. The EOR methods may be classified into the following kinds: injection of chemical solutions, injection of solvents and thermal methods. The chemical fluids most commonly injected are polymers, surfactants, micellar solutions, etc. Displacement of oil by any of these fluids involves complex physico-chemical processes of interphase mass transfer, phase transitions and transport properties changes. These processes can be divided into two main categories: thermodynamical and hydrodynamical ones. They occur simultaneously during the displacement, and are coupled in the modern mathematical models of EOR. The model for one-dimensional displacement of oil by polymer solutions is analyzed in this paper. The Courant number is fixed, and we compare the results of different runs of a numerical simulator with the analytical solution of this problem. Each run corresponds to a different spatial discretization. (author)
Numerical calculation of the cross section by the solution of the wave equation
International Nuclear Information System (INIS)
Drewko, J.
1982-01-01
A numerical method of solving of the wave equation is described for chosen vibrational eigenfunctions. A prepared program calculates the total cross sections for the resonant vibrational excitation for diatomic molecules on the basis of introduced molecular data. (author)
Numerical solution for multi-term fractional (arbitrary) orders differential equations
El-Sayed, A. M. A.; El-Mesiry, A. E. M.; El-Saka, H. A. A.
2004-01-01
Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation. Some results concerning the existence and uniqueness have been also obtained.
Numerical solution of neutral functional-differential equations with proportional delays
Directory of Open Access Journals (Sweden)
Mehmet Giyas Sakar
2017-07-01
Full Text Available In this paper, homotopy analysis method is improved with optimal determination of auxiliary parameter by use of residual error function for solving neutral functional-differential equations (NFDEs with proportional delays. Convergence analysis and error estimate of method are given. Some numerical examples are solved and comparisons are made with the existing results. The numerical results show that the homotopy analysis method with residual error function is very effective and simple.
Cubic spline numerical solution of an ablation problem with convective backface cooling
Lin, S.; Wang, P.; Kahawita, R.
1984-08-01
An implicit numerical technique using cubic splines is presented for solving an ablation problem on a thin wall with convective cooling. A non-uniform computational mesh with 6 grid points has been used for the numerical integration. The method has been found to be computationally efficient, providing for the care under consideration of an overall error of about 1 percent. The results obtained indicate that the convective cooling is an important factor in reducing the ablation thickness.
On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations
Directory of Open Access Journals (Sweden)
H. Montazeri
2012-01-01
Full Text Available We consider a system of nonlinear equations F(x=0. A new iterative method for solving this problem numerically is suggested. The analytical discussions of the method are provided to reveal its sixth order of convergence. A discussion on the efficiency index of the contribution with comparison to the other iterative methods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica.
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II
International Nuclear Information System (INIS)
Larsen, E.W.; Morel, J.E.
1989-01-01
In a recent article (Larsen, Morel, and Miller, J. Comput. Phys. 69, 283 (1987)), a theoretical method is described for assessing the accuracy of transport differencing schemes in highly scattering media with optically thick spatial meshes. In the present article, this method is extended to enable one to determine the accuracy of such schemes in the presence of numerically unresolved boundary layers. Numerical results are presented that demonstrate the validity and accuracy of our analysis. copyright 1989 Academic Press, Inc
International Nuclear Information System (INIS)
Mahanthesh, B.; Gireesha, B.J.; Gorla, R.S. Reddy; Abbasi, F.M.; Shehzad, S.A.
2016-01-01
Numerical solutions of three-dimensional flow over a non-linear stretching surface are developed in this article. An electrically conducting flow of viscous nanoliquid is considered. Heat transfer phenomenon is accounted under thermal radiation, Joule heating and viscous dissipation effects. We considered the variable heat flux condition at the surface of sheet. The governing mathematical equations are reduced to nonlinear ordinary differential systems through suitable dimensionless variables. A well-known shooting technique is implemented to obtain the results of dimensionless velocities and temperature. The obtained results are plotted for multiple values of pertinent parameters to discuss the salient features of these parameters on fluid velocity and temperature. The expressions of skin-friction coefficient and Nusselt number are computed and analyzed comprehensively through numerical values. A comparison of present results with the previous results in absence of nanoparticle volume fraction, mixed convection and magnetic field is computed and an excellent agreement noticed. We also computed the results for both linear and non-linear stretching sheet cases. - Highlights: • Hydromagnetic flow of nanofluid over a bidirectional non-linear stretching surface is examined. • Cu, Al 2 O3 and TiO 2 types nanoparticles are taken into account. • Numerical solutions have been computed and addressed. • The values of skin-friction and Nusselt number are presented.
Energy Technology Data Exchange (ETDEWEB)
Mahanthesh, B., E-mail: bmanths@gmail.com [Department of Mathematics, AIMS Institutes, Peenya, 560058 Bangalore (India); Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, 577451 Shimoga, Karnataka (India); Gireesha, B.J., E-mail: bjgireesu@rediffmail.com [Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, 577451 Shimoga, Karnataka (India); Department of Mechanical Engineering, Cleveland State University, Cleveland, OH (United States); Gorla, R.S. Reddy, E-mail: r.gorla@csuohio.edu [Department of Mechanical Engineering, Cleveland State University, Cleveland, OH (United States); Abbasi, F.M., E-mail: abbasisarkar@gmail.com [Department of Mathematics, Comsats Institute of Information Technology, Islamabad 44000 (Pakistan); Shehzad, S.A., E-mail: ali_qau70@yahoo.com [Department of Mathematics, Comsats Institute of Information Technology, Sahiwal 57000 (Pakistan)
2016-11-01
Numerical solutions of three-dimensional flow over a non-linear stretching surface are developed in this article. An electrically conducting flow of viscous nanoliquid is considered. Heat transfer phenomenon is accounted under thermal radiation, Joule heating and viscous dissipation effects. We considered the variable heat flux condition at the surface of sheet. The governing mathematical equations are reduced to nonlinear ordinary differential systems through suitable dimensionless variables. A well-known shooting technique is implemented to obtain the results of dimensionless velocities and temperature. The obtained results are plotted for multiple values of pertinent parameters to discuss the salient features of these parameters on fluid velocity and temperature. The expressions of skin-friction coefficient and Nusselt number are computed and analyzed comprehensively through numerical values. A comparison of present results with the previous results in absence of nanoparticle volume fraction, mixed convection and magnetic field is computed and an excellent agreement noticed. We also computed the results for both linear and non-linear stretching sheet cases. - Highlights: • Hydromagnetic flow of nanofluid over a bidirectional non-linear stretching surface is examined. • Cu, Al{sub 2}O3 and TiO{sub 2} types nanoparticles are taken into account. • Numerical solutions have been computed and addressed. • The values of skin-friction and Nusselt number are presented.
On the numerical solution of the Gross–Pitaevskii equation | Laoye ...
African Journals Online (AJOL)
The Gross–Pitaevskii equation is solved using an approach developed for the solution of the Bogoliubov–de Gennes equations for type II superconductivity. The solution is compared with others in the literature and is shown to be easily adapted to the study of an isolated vortex recently discovered in Bose-Einstein ...
International Nuclear Information System (INIS)
Valdes Parra, J.J.
1986-01-01
One of the main problems in reactor physics is to determine the neutron distribution in reactor core, since knowing that, it is possible to calculate the rapidity of occurrence of different nuclear reaction inside the reactor core. Within different theories existing in nuclear reactor physics, is neutron transport the one in which equation who govern the exact behavior of neutronic distribution are developed even inside the proper neutron transport theory, there exist different methods of solution which are approximations to exact solution; still more, with the purpose to reach a more precise solution, the majority of methods have been approached to the obtention of solutions in numerical form with the aim of take the advantages of modern computers, and for this reason a great deal of effort is dedicated to numerical solution of the equations of neutron transport. In agreement with the above mentioned, in this work has been developed a computer program which uses a relatively new techniques known as 'acceleration of synthetic diffusion' which has been applied to solve the neutron transport equation with 'classical schemes of spatial integration' obtaining results with a smaller quantity of interactions, if they compare to done without using such equation (Author)
International Nuclear Information System (INIS)
Caraballo, T.; Kloeden, P.E.
2006-01-01
Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions
Directory of Open Access Journals (Sweden)
Fukang Yin
2013-01-01
Full Text Available A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs. The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs. The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
International Nuclear Information System (INIS)
Kotler, Z.; Neria, E.; Nitzan, A.
1991-01-01
The use of the time-dependent self-consistent field approximation (TDSCF) in the numerical solution of quantum curve crossing and tunneling dynamical problems is investigated. Particular emphasis is given to multiconfiguration TDSCF (MCTDSCF) approximations, which are shown to perform considerably better with only a small increase in computational effort. We investigate a number of simple models in which a 'system' characterized by two electronic potential surfaces evolves while interacting with a 'bath' mode described by an harmonic oscillator, and compare exact numerical solutions to one- and two-configuration TDSCF approximations. We also introduce and investigate a semiclassical approximation in which the 'bath' mode is described by semiclassical wavepackets (one for each electronic state) and show that for all models investigated this scheme works very well in comparison with the fully quantum MCTDSCF approximation. This provides a potentially very useful method to simulate strongly quantum systems coupled to an essentially classical environment. (orig.)
International Nuclear Information System (INIS)
Dinh Nho Hao; Nguyen Trung Thanh; Sahli, Hichem
2008-01-01
In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by aids of an adjoint problem and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.
Directory of Open Access Journals (Sweden)
Vincenzo Maria De Benedictis
2016-05-01
Full Text Available The Mark–Houwink–Sakurada (MHS equation allows for estimation of rheological properties, if the molecular weight is known along with good understanding of the polymer conformation. The intrinsic viscosity of a polymer solution is related to the polymer molecular weight according to the MHS equation, where the value of the constants is related to the specific solvent and its concentration. However, MHS constants do not account for other characteristics of the polymeric solutions, i.e., Deacetilation Degree (DD when the solute is chitosan. In this paper, the degradation of chitosan in different acidic environments by thermal treatment is addressed. In particular, two different solutions are investigated (used as solvent acetic or hydrochloric acid with different concentrations used for the preparation of chitosan solutions. The samples were treated at different temperatures (4, 30, and 80 °C and time points (3, 6 and 24 h. Rheological, Gel Permeation Chromatography (GPC, Fourier Transform Infrared Spectroscopy (FT-IR, Differential Scanning Calorimetry (DSC and Thermal Gravimetric Analyses (TGA were performed in order to assess the degradation rate of the polymer backbones. Measured values of molecular weight have been integrated in the simulation of the batch degradation of chitosan solutions for evaluating MHS coefficients to be compared with their corresponding experimental values. Evaluating the relationship between the different parameters used in the preparation of chitosan solutions (e.g., temperature, time, acid type and concentration, and their contribution to the degradation of chitosan backbone, it is important to have a mathematical frame that could account for phenomena involved in polymer degradation that go beyond the solvent-solute combination. Therefore, the goal of the present work is to propose an integration of MHS coefficients for chitosan solutions that contemplate a deacetylation degree for chitosan systems or a more
International Nuclear Information System (INIS)
Koeppel, T.; Harvey, M.
1984-06-01
A new numerical method is applied to solving the equations of motion of the Friedberg-Lee Soliton model for both ground and spherically symmetric excited states. General results have been obtained over a wide range of parameters. Critical coupling constants and critical particle numbers have been determined below which soliton solutions cease to exist. The static properties of the proton are considered to show that as presently formulated the model fails to fit all experimental data for any set of parameters
Markou, A. A.; Manolis, G. D.
2018-03-01
Numerical methods for the solution of dynamical problems in engineering go back to 1950. The most famous and widely-used time stepping algorithm was developed by Newmark in 1959. In the present study, for the first time, the Newmark algorithm is developed for the case of the trilinear hysteretic model, a model that was used to describe the shear behaviour of high damping rubber bearings. This model is calibrated against free-vibration field tests implemented on a hybrid base isolated building, namely the Solarino project in Italy, as well as against laboratory experiments. A single-degree-of-freedom system is used to describe the behaviour of a low-rise building isolated with a hybrid system comprising high damping rubber bearings and low friction sliding bearings. The behaviour of the high damping rubber bearings is simulated by the trilinear hysteretic model, while the description of the behaviour of the low friction sliding bearings is modeled by a linear Coulomb friction model. In order to prove the effectiveness of the numerical method we compare the analytically solved trilinear hysteretic model calibrated from free-vibration field tests (Solarino project) against the same model solved with the Newmark method with Netwon-Raphson iteration. Almost perfect agreement is observed between the semi-analytical solution and the fully numerical solution with Newmark's time integration algorithm. This will allow for extension of the trilinear mechanical models to bidirectional horizontal motion, to time-varying vertical loads, to multi-degree-of-freedom-systems, as well to generalized models connected in parallel, where only numerical solutions are possible.
Directory of Open Access Journals (Sweden)
Markou A.A.
2018-03-01
Full Text Available Numerical methods for the solution of dynamical problems in engineering go back to 1950. The most famous and widely-used time stepping algorithm was developed by Newmark in 1959. In the present study, for the first time, the Newmark algorithm is developed for the case of the trilinear hysteretic model, a model that was used to describe the shear behaviour of high damping rubber bearings. This model is calibrated against free-vibration field tests implemented on a hybrid base isolated building, namely the Solarino project in Italy, as well as against laboratory experiments. A single-degree-of-freedom system is used to describe the behaviour of a low-rise building isolated with a hybrid system comprising high damping rubber bearings and low friction sliding bearings. The behaviour of the high damping rubber bearings is simulated by the trilinear hysteretic model, while the description of the behaviour of the low friction sliding bearings is modeled by a linear Coulomb friction model. In order to prove the effectiveness of the numerical method we compare the analytically solved trilinear hysteretic model calibrated from free-vibration field tests (Solarino project against the same model solved with the Newmark method with Netwon-Raphson iteration. Almost perfect agreement is observed between the semi-analytical solution and the fully numerical solution with Newmark’s time integration algorithm. This will allow for extension of the trilinear mechanical models to bidirectional horizontal motion, to time-varying vertical loads, to multi-degree-of-freedom-systems, as well to generalized models connected in parallel, where only numerical solutions are possible.
International Nuclear Information System (INIS)
Sarmiento, G.S.; Laura, P.A.A.
1979-01-01
Domains of complicated boundary shape are of great practical importance in several fields of technology and applied science; e.g. solid propellant rocket grains, electromagnetic and acoustic waveguides, and certain elements used in nuclear engineering. The technical literature contains very few comparative studies of analytical and numerical solutions when dealing with such rather complex geometries. The present study constitutes an effort in that direction. (Auth.)
Atlabachew, Abunu; Shu, Longcang; Wu, Peipeng; Zhang, Yongjie; Xu, Yang
2018-03-01
This laboratory study improves the understanding of the impacts of horizontal hydraulic gradient, artificial recharge, and groundwater pumping on solute transport through aquifers. Nine experiments and numerical simulations were carried out using a sand tank. The variable-density groundwater flow and sodium chloride transport were simulated using the three-dimensional numerical model SEAWAT. Numerical modelling results successfully reproduced heads and concentrations observed in the sand tank. A higher horizontal hydraulic gradient enhanced the migration of sodium chloride, particularly in the groundwater flow direction. The application of constant artificial recharge increased the spread of the sodium chloride plume in both the longitudinal and lateral directions. In addition, groundwater pumping accelerated spreading of the sodium chloride plume towards the pumping well. Both higher hydraulic gradient and pumping rate generated oval-shaped plumes in the horizontal plane. However, the artificial recharge process produced stretched plumes. These effects of artificial recharge and groundwater pumping were greater under higher hydraulic gradient. The concentration breakthrough curves indicated that emerging solutions never attained the concentration of the originally injected solution. This is probably because of sorption of sodium chloride onto the silica sand and/or the exchange of sodium chloride between the mobile and immobile liquid domains. The fingering and protruding plume shapes in the numerical models constitute instability zones produced by buoyancy-driven flow. Overall, the results have substantiated the influences of hydraulic gradient, boundary condition, artificial recharge, pumping rate and density differences on solute transport through a homogeneous unconfined aquifer. The implications of these findings are important for managing liquid wastes.
Schneider, André; Lin, Zhongbing; Sterckeman, Thibault; Nguyen, Christophe
2018-04-01
The dissociation of metal complexes in the soil solution can increase the availability of metals for root uptake. When it is accounted for in models of bioavailability of soil metals, the number of partial differential equations (PDEs) increases and the computation time to numerically solve these equations may be problematic when a large number of simulations are required, for example for sensitivity analyses or when considering root architecture. This work presents analytical solutions for the set of PDEs describing the bioavailability of soil metals including the kinetics of complexation for three scenarios where the metal complex in solution was fully inert, fully labile, or partially labile. The analytical solutions are only valid i) at steady-state when the PDEs become ordinary differential equations, the transient phase being not covered, ii) when diffusion is the major mechanism of transport and therefore, when convection is negligible, iii) when there is no between-root competition. The formulation of the analytical solutions is for cylindrical geometry but the solutions rely on the spread of the depletion profile around the root, which was modelled assuming a planar geometry. The analytical solutions were evaluated by comparison with the corresponding PDEs for cadmium in the case of the French agricultural soils. Provided that convection was much lower than diffusion (Péclet's number<0.02), the cumulative uptakes calculated from the analytic solutions were in very good agreement with those calculated from the PDEs, even in the case of a partially labile complex. The analytic solutions can be used instead of the PDEs to predict root uptake of metals. The analytic solutions were also used to build an indicator of the contribution of a complex to the uptake of the metal by roots, which can be helpful to predict the effect of soluble organic matter on the bioavailability of soil metals. Copyright © 2017 Elsevier B.V. All rights reserved.
Numerical solution of special ultra-relativistic Euler equations using central upwind scheme
Ghaffar, Tayabia; Yousaf, Muhammad; Qamar, Shamsul
2018-06-01
This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.
Chew, J. V. L.; Sulaiman, J.
2017-09-01
Partial differential equations that are used in describing the nonlinear heat and mass transfer phenomena are difficult to be solved. For the case where the exact solution is difficult to be obtained, it is necessary to use a numerical procedure such as the finite difference method to solve a particular partial differential equation. In term of numerical procedure, a particular method can be considered as an efficient method if the method can give an approximate solution within the specified error with the least computational complexity. Throughout this paper, the two-dimensional Porous Medium Equation (2D PME) is discretized by using the implicit finite difference scheme to construct the corresponding approximation equation. Then this approximation equation yields a large-sized and sparse nonlinear system. By using the Newton method to linearize the nonlinear system, this paper deals with the application of the Four-Point Newton-EGSOR (4NEGSOR) iterative method for solving the 2D PMEs. In addition to that, the efficiency of the 4NEGSOR iterative method is studied by solving three examples of the problems. Based on the comparative analysis, the Newton-Gauss-Seidel (NGS) and the Newton-SOR (NSOR) iterative methods are also considered. The numerical findings show that the 4NEGSOR method is superior to the NGS and the NSOR methods in terms of the number of iterations to get the converged solutions, the time of computation and the maximum absolute errors produced by the methods.
Ikeguchi, Mitsunori; Doi, Junta
1995-09-01
The Ornstein-Zernike integral equation (OZ equation) has been used to evaluate the distribution function of solvents around solutes, but its numerical solution is difficult for molecules with a complicated shape. This paper proposes a numerical method to directly solve the OZ equation by introducing the 3D lattice. The method employs no approximation the reference interaction site model (RISM) equation employed. The method enables one to obtain the spatial distribution of spherical solvents around solutes with an arbitrary shape. Numerical accuracy is sufficient when the grid-spacing is less than 0.5 Å for solvent water. The spatial water distribution around a propane molecule is demonstrated as an example of a nonspherical hydrophobic molecule using iso-value surfaces. The water model proposed by Pratt and Chandler is used. The distribution agrees with the molecular dynamics simulation. The distribution increases offshore molecular concavities. The spatial distribution of water around 5α-cholest-2-ene (C27H46) is visualized using computer graphics techniques and a similar trend is observed.
International Nuclear Information System (INIS)
Quang A, Dang; Hai, Truong Ha
2014-01-01
Very recently in the work S imple Iterative Method for Solving Problems for Plates with Partial Internal Supports, Journal of Engineering Mathematics, DOI: 10.1007/s10665-013-9652-7 (in press) , we proposed a numerical method for solving some problems of plates on one and two line partial internal supports (LPIS). In the essence they are problems with strongly mixed boundary conditions for biharmonic equation. Using this method we reduced the problems to a sequence of boundary value problems for the Poisson equation with weakly mixed boundary conditions, which are easily solved numerically. The advantages of the method over other ones were shown. In this paper we apply the method to plates on internal supports of more complicated configurations. Namely, we consider the case of three LPIS and the case of the cross support. The convergence of the method is established theoretically and its efficiency is confirmed on numerical experiments
International Nuclear Information System (INIS)
Talamo, Alberto
2013-01-01
This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps
Energy Technology Data Exchange (ETDEWEB)
Talamo, Alberto, E-mail: alby@anl.gov [Nuclear Engineering Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439 (United States)
2013-05-01
This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.
International Nuclear Information System (INIS)
Imhof, Armando Luis; Calvo, Carlos Adolfo; Moyano, Amalia; Sanchez, Manuel
2015-01-01
A determined curve path is followed by the propagation of seismic waves generated in emitters and detected in receivers by the principle of minimum time of Fermat. An ordinary differential equation is derived from the application of the calculation of variations. Due to the compaction of the terrain, the speed usually increases with depth. The experimental laws for each soil have led to this variation leading to a numerical resolution. The adjustment of experimental speed data by an exponential function; the analytical integration of the differential equation and the numerical determination of the integration constants are studied. A geophysical method such as up-hole or down-hole has determined the experimental data. Its main application is centered in the validation of numerical models of curved trajectories. Then time of first arrivals through tomographic algorithms for detection and modeling of anomalies in the first 12 m depth. (author) [es
Numerical solution of two-dimensional non-linear partial differential ...
African Journals Online (AJOL)
linear partial differential equations using a hybrid method. The solution technique involves discritizing the non-linear system of partial differential equations (PDEs) to obtain a corresponding nonlinear system of algebraic difference equations to be ...
Analytical and numerical solutions of the Schrödinger–KdV equation
Indian Academy of Sciences (India)
solitary wave ansatz method is used to carry out the integration of the ..... Exact rational travelling wave solutions of SKdV equation obtained by the ...... Air Force Office of Scientific Research (AFOSR) under the award number: W54428-RT-.
Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks
Directory of Open Access Journals (Sweden)
Raheleh Jafari
2017-01-01
Full Text Available The uncertain nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs by incorporating the fuzzy set theory. The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by two types of Bernstein neural networks. Here, the uncertainties are in the sense of Z-numbers. Initially the FDE is transformed into four ordinary differential equations (ODEs with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z- number variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs based on Z-numbers.
Exponential and Bessel fitting methods for the numerical solution of the Schroedinger equation
International Nuclear Information System (INIS)
Raptis, A.D.; Cash, J.R.
1987-01-01
A new method is developed for the numerical integration of the one dimensional radial Schroedinger equation. This method involves using different integration formulae in different parts of the range of integration rather than using the same integration formula throughout. Two new integration formulae are derived, one which integrates Bessel and Neumann functions exactly and another which exactly integrates certain exponential functions. It is shown that, for large r, these new formulae are much more accurate than standard integration methods for the Schroedinger equation. The benefit of using this new approach is demonstrated by considering some numerical examples based on the Lennard-Jones potential. (orig.)
Sweilam, N. H.; Abou Hasan, M. M.
2017-05-01
In this paper, the weighted-average non-standard finite-difference (WANSFD) method is used to study numerically the general time-fractional nonlinear, one-dimensional problem of thermoelasticity. This model contains the standard system arising in thermoelasticity as a special case. The stability of the proposed method is analyzed by a procedure akin to the standard John von Neumann technique. Moreover, the accuracy of the proposed scheme is proved. Numerical results are presented graphically, which reveal that the WANSFD method is easy to implement, effective and convenient for solving the proposed system. The proposed method could also be easily extended to solve other systems of fractional partial differential equations.
On the efficient numerical solution of lattice systems with low-order couplings
International Nuclear Information System (INIS)
Ammon, A.; Genz, A.; Hartung, T.; Jansen, K.; Volmer, J.; Leoevey, H.
2015-10-01
We apply the Quasi Monte Carlo (QMC) and recursive numerical integration methods to evaluate the Euclidean, discretized time path-integral for the quantum mechanical anharmonic oscillator and a topological quantum mechanical rotor model. For the anharmonic oscillator both methods outperform standard Markov Chain Monte Carlo methods and show a significantly improved error scaling. For the quantum mechanical rotor we could, however, not find a successful way employing QMC. On the other hand, the recursive numerical integration method works extremely well for this model and shows an at least exponentially fast error scaling.
Numerical Solution of Diffusion Models in Biomedical Imaging on Multicore Processors
Directory of Open Access Journals (Sweden)
Luisa D'Amore
2011-01-01
Full Text Available In this paper, we consider nonlinear partial differential equations (PDEs of diffusion/advection type underlying most problems in image analysis. As case study, we address the segmentation of medical structures. We perform a comparative study of numerical algorithms arising from using the semi-implicit and the fully implicit discretization schemes. Comparison criteria take into account both the accuracy and the efficiency of the algorithms. As measure of accuracy, we consider the Hausdorff distance and the residuals of numerical solvers, while as measure of efficiency we consider convergence history, execution time, speedup, and parallel efficiency. This analysis is carried out in a multicore-based parallel computing environment.
International Nuclear Information System (INIS)
Lim, S.C.; Lee, K.J.
1993-01-01
The Galerkin finite element method is used to solve the problem of one-dimensional, vertical flow of water and mass transport of conservative-nonconservative solutes in unsaturated porous media. Numerical approximations based on different forms of the governing equation, although they are equivalent in continuous forms, can result in remarkably different solutions in an unsaturated flow problem. Solutions given by a simple Galerkin method based on the h-based Richards equation yield a large mass balance error and an underestimation of the infiltration depth. With the employment of the ROMV (restoration of main variable) concept in the discretization step, the mass conservative numerical solution algorithm for water flow has been derived. The resulting computational schemes for water flow and mass transport are applied to sandy soil. The ROMV method shows good mass conservation in water flow analysis, whereas it seems to have a minor effect on mass transport. However, it may relax the time-step size restriction and so ensure an improved calculation output. (author)
Cai, Haibing; Xu, Liuxun; Yang, Yugui; Li, Longqi
2018-05-01
Artificial liquid nitrogen freezing technology is widely used in urban underground engineering due to its technical advantages, such as simple freezing system, high freezing speed, low freezing temperature, high strength of frozen soil, and absence of pollution. However, technical difficulties such as undefined range of liquid nitrogen freezing and thickness of frozen wall gradually emerge during the application process. Thus, the analytical solution of the freezing-temperature field of a single pipe is established considering the freezing temperature of soil and the constant temperature of freezing pipe wall. This solution is then applied in a liquid nitrogen freezing project. Calculation results show that the radius of freezing front of liquid nitrogen is proportional to the square root of freezing time. The radius of the freezing front also decreases with decreased the freezing temperature, and the temperature gradient of soil decreases with increased distance from the freezing pipe. The radius of cooling zone in the unfrozen area is approximately four times the radius of the freezing front. Meanwhile, the numerical simulation of the liquid nitrogen freezing-temperature field of a single pipe is conducted using the Abaqus finite-element program. Results show that the numerical simulation of soil temperature distribution law well agrees with the analytical solution, further verifies the reliability of the established analytical solution of the liquid nitrogen freezing-temperature field of a single pipe.
A mathematical model and numerical solution of interface problems for steady state heat conduction
Directory of Open Access Journals (Sweden)
Z. Muradoglu Seyidmamedov
2006-01-01
(isolation Ωδ tends to zero. For each case, the local truncation errors of the used conservative finite difference scheme are estimated on the nonuniform grid. A fast direct solver has been applied for the interface problems with piecewise constant but discontinuous coefficient k=k(x. The presented numerical results illustrate high accuracy and show applicability of the given approach.
DEFF Research Database (Denmark)
Lee, Jonghyun; Rolle, Massimo; Kitanidis, Peter K.
2018-01-01
Most recent research on hydrodynamic dispersion in porous media has focused on whole-domain dispersion while other research is largely on laboratory-scale dispersion. This work focuses on the contribution of a single block in a numerical model to dispersion. Variability of fluid velocity and conc...
Global communication schemes for the numerical solution of high-dimensional PDEs
DEFF Research Database (Denmark)
Hupp, Philipp; Heene, Mario; Jacob, Riko
2016-01-01
The numerical treatment of high-dimensional partial differential equations is among the most compute-hungry problems and in urgent need for current and future high-performance computing (HPC) systems. It is thus also facing the grand challenges of exascale computing such as the requirement...
Direct numerical solution of Poisson's equation in cylindrical (r, z) coordinates
International Nuclear Information System (INIS)
Chao, E.H.; Paul, S.F.; Davidson, R.C.; Fine, K.S.
1997-01-01
A direct solver method is developed for solving Poisson's equation numerically for the electrostatic potential φ(r,z) in a cylindrical region (r wall , 0 wall , z) are specified, and ∂φ/∂z = 0 at the axial boundaries (z = 0, L)
Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method
Krasnov, M. M.; Kuchugov, P. A.; E Ladonkina, M.; E Lutsky, A.; Tishkin, V. F.
2017-02-01
Detailed unstructured grids and numerical methods of high accuracy are frequently used in the numerical simulation of gasdynamic flows in areas with complex geometry. Galerkin method with discontinuous basis functions or Discontinuous Galerkin Method (DGM) works well in dealing with such problems. This approach offers a number of advantages inherent to both finite-element and finite-difference approximations. Moreover, the present paper shows that DGM schemes can be viewed as Godunov method extension to piecewise-polynomial functions. As is known, DGM involves significant computational complexity, and this brings up the question of ensuring the most effective use of all the computational capacity available. In order to speed up the calculations, operator programming method has been applied while creating the computational module. This approach makes possible compact encoding of mathematical formulas and facilitates the porting of programs to parallel architectures, such as NVidia CUDA and Intel Xeon Phi. With the software package, based on DGM, numerical simulations of supersonic flow past solid bodies has been carried out. The numerical results are in good agreement with the experimental ones.
A three–step discretization scheme for direct numerical solution of ...
African Journals Online (AJOL)
In this paper, a three-step discretization (numerical) formula is developed for direct integration of second-order initial value problems in ordinary differential equations. The development of the method and analysis of its basic properties adopt Taylor series expansion and Dahlquist stability test methods. The results show that ...
A numerical solution for the multi-server queue with hyper-exponential service times
de Smit, J.H.A.
1983-01-01
In this paper we present a numerical method for the queue GI/H2/s, which is based on general results for GI/Hm/s. We give a complete description of the algorithm which yields exact results for the steady distributions of the actual waiting time, the virtual waiting time and the number of customers
An efficient numerical target strength prediction model: Validation against analysis solutions
Fillinger, L.; Nijhof, M.J.J.; Jong, C.A.F. de
2014-01-01
A decade ago, TNO developed RASP (Rapid Acoustic Signature Prediction), a numerical model for the prediction of the target strength of immersed underwater objects. The model is based on Kirchhoff diffraction theory. It is currently being improved to model refraction, angle dependent reflection and
International Nuclear Information System (INIS)
Abourida, B.; Hasnaoui, M.
2005-01-01
Laminar natural convection in an infinite horizontal channel heated periodically from below and provided with thin adiabatic partitions on its lower wall, is investigated numerically. The effect of these partitions on the multiplicity of solutions and heat transfer characteristics in the computational domain is studied. The parameters of the study are the Rayleigh number (10 2 Ra 4.9 x 10 6 ) and the height of the partitions (0 B = h'/H' 1/2). The results obtained in the case of air (Pr = 0.72) as working fluid show that depending on the governing parameters, the existence of multiple solutions is possible. Important differences in terms of heat transfer are observed between two different solutions
International Nuclear Information System (INIS)
Velloso, P.A.; Galeao, A.C.
1989-05-01
This paper deals with nonlinear vibrations of pipes subjected to non-conservative loads. Periodic solutions of these problems are determined using a variational approach based on Hamilton's Principle combined with a Fourier series expansion to describe the displacement field time dependence. A finite element model which utilizes Hemite's cubic interpolation for both axial and transversal displacement amplitudes is used. This model is applied to the problem of a pipe subjected to a tangential and a normal follower force. The numerical results obtained with this model are compared with the corespondent solutions determined using a total lagrangian description for the Principle of Virtual Work, coupled with Newmark's step-by-step integration procedure. It is shown that for small to moderate displacement amplitudes the one-term Fourier series approximation compares fairly well with the predicted solution. For large displacements as least a two-term approximation should be utilized [pt
International Nuclear Information System (INIS)
Shin, Y.W.; Wiedermann, A.H.
1979-10-01
A solution method is presented for transient, homogeneous, equilibrium, two-phase flows of a single-component fluid in one space dimension. The method combines a direct finite-difference procedure and the method of characteristics. The finite-difference procedure solves the interior points of the computing domain; the boundary information is provided by a separate procedure based on the characteristics theory. The solution procedure for boundary points requires information in addition to the physical boundary conditions. This additional information is obtained by a new procedure involving integration of characteristics in the hodograph plane. Sample problems involving various combinations of basic boundary types are calculated for two-phase water/steam mixtures and single-phase nitrogen gas, and compared with independent method-of-characteristics solutions using very fine characteristic mesh. In all cases, excellent agreement is demonstrated
International Nuclear Information System (INIS)
Savovic, S.; Djordjevich, A.; Ristic, G.
2012-01-01
A theoretical evaluation of the properties and processes affecting the radon transport from subsurface soil into buildings is presented in this work. The solution of the relevant transport equation is obtained using the explicit finite difference method (EFDM). Results are compared with analytical steady-state solution reported in the literature. Good agreement is found. It is shown that EFDM is effective and accurate for solving the equation that describes radon diffusion, advection and decay during its transport from subsurface to buildings, which is especially important when arbitrary initial and boundary conditions are required. (authors)
NUMERICAL SOLUTION OF SINGULAR INVERSE NODAL PROBLEM BY USING CHEBYSHEV POLYNOMIALS
NEAMATY, ABDOLALI; YILMAZ, EMRAH; AKBARPOOR, SHAHRBANOO; DABBAGHIAN, ABDOLHADI
2017-01-01
In this study, we consider Sturm-Liouville problem in two cases: the first case having no singularity and the second case having a singularity at zero. Then, we calculate the eigenvalues and the nodal points and present the uniqueness theorem for the solution of the inverse problem by using a dense subset of the nodal points in two given cases. Also, we use Chebyshev polynomials of the first kind for calculating the approximate solution of the inverse nodal problem in these cases. Finally, we...
Numerical-solution package for transient two-phase-flow equations
International Nuclear Information System (INIS)
Mahaffy, J.H.; Liles, D.
1982-01-01
The methods presented have proven to be extremely reliable tools for reactor safety analysis. They can handle a wide range of fluid conditions and time scales with minimal failures, while maintaining time step sizes well above those possible with most other techniques. This robustness is due not only to the finite-difference equations themselves, but also to the choice of solution technique, because poorly chosen iterative solution procedures often require a limit on the time-step size for proper convergence of the iteration
Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework
Directory of Open Access Journals (Sweden)
H. R. Marzban
2016-01-01
Full Text Available An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique.
Directory of Open Access Journals (Sweden)
Carlos Humberto Galeano Urueña
2009-05-01
Full Text Available This article describes the streamline upwind Petrov-Galerkin (SUPG method as being a stabilisation technique for resolving the diffusion-advection-reaction equation by finite elements. The first part of this article has a short analysis of the importance of this type of differential equation in modelling physical phenomena in multiple fields. A one-dimensional description of the SUPG me- thod is then given to extend this basis to two and three dimensions. The outcome of a strongly advective and a high numerical complexity experiment is presented. The results show how the version of the implemented SUPG technique allowed stabilised approaches in space, even for high Peclet numbers. Additional graphs of the numerical experiments presented here can be downloaded from www.gnum.unal.edu.co.
Scott, James R.; Atassi, Hafiz M.
1990-01-01
A linearized unsteady aerodynamic analysis is presented for unsteady, subsonic vortical flows around lifting airfoils. The analysis fully accounts for the distortion effects of the nonuniform mean flow on the imposed vortical disturbances. A frequency domain numerical scheme which implements this linearized approach is described, and numerical results are presented for a large variety of flow configurations. The results demonstrate the effects of airfoil thickness, angle of attack, camber, and Mach number on the unsteady lift and moment of airfoils subjected to periodic vortical gusts. The results show that mean flow distortion can have a very strong effect on the airfoil unsteady response, and that the effect depends strongly upon the reduced frequency, Mach number, and gust wave numbers.
Two-dimensional numerical modeling and solution of convection heat transfer in turbulent He II
Zhang, Burt X.; Karr, Gerald R.
1991-01-01
Numerical schemes are employed to investigate heat transfer in the turbulent flow of He II. FEM is used to solve a set of equations governing the heat transfer and hydrodynamics of He II in the turbulent regime. Numerical results are compared with available experimental data and interpreted in terms of conventional heat transfer parameters such as the Prandtl number, the Peclet number, and the Nusselt number. Within the prescribed Reynolds number domain, the Gorter-Mellink thermal counterflow mechanism becomes less significant, and He II acts like an ordinary fluid. The convection heat transfer characteristics of He II in the highly turbulent regime can be successfully described by using the conventional turbulence and heat transfer theories.
Numerical solution of critical state in superconductivity by finite element software
Energy Technology Data Exchange (ETDEWEB)
Hong, Z; Campbell, A M; Coombs, T A [Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ (United Kingdom)
2006-12-15
A numerical method is proposed to analyse the electromagnetic behaviour of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current. The E-J constitutive law together with an H-formulation is used to calculate the current distribution and electromagnetic fields in HTSCs, and the magnetization of HTSCs; then the forces in the interaction between the electromagnet and the superconductor and the AC loss of the superconducting cable can be obtained. This numerical method is based on solving the partial differential equations time dependently and is adapted to the commercial finite element software Comsol Multiphysics 3.2. The advantage of this method is to make the modelling of the superconductivity simple, flexible and extendable.
Numerical solution of recirculating flow by a simple finite element recursion relation
Energy Technology Data Exchange (ETDEWEB)
Pepper, D W; Cooper, R E
1980-01-01
A time-split finite element recursion relation, based on linear basis functions, is used to solve the two-dimensional equations of motion. Recirculating flow in a rectangular cavity and free convective flow in an enclosed container are analyzed. The relation has the advantage of finite element accuracy and finite difference speed and simplicity. Incorporating dissipation parameters in the functionals decreases numerical dispersion and improves phase lag.
Numerical solution of multi groups point kinetic equations by simulink toolbox of Matlab software
International Nuclear Information System (INIS)
Hadad, K.; Mohamadi, A.; Sabet, H.; Ayobian, N.; Khani, M.
2004-01-01
The simulink toolbox of Matlab Software was employed to solve the point kinetics equation with six group delayed neutrons. The method of Adams-Bash ford showed a good convergence in solving the system of simultaneous equations and the obtained results showed good agreements with other numerical schemes. The flexibility of the package in changing the system parameters and the user friendly interface makes this approach a reliable educational package in revealing the affects of reactivity changes on power incursions
International Nuclear Information System (INIS)
Krastev, A.; Nedialkov, J.
1983-01-01
A method for numerical solving the equation of the computerized tomography is proposed. The method, is based on the Cormack's formulae and is applied for studying media with central symmetry. The generalization for the nonsymmetric case does not lead to complications. The method is applied for the investigation of the density distribution of the Earth, the other planets and the Sun by means of neutrino experiments
Czech Academy of Sciences Publication Activity Database
Pokorný, Vladislav; Žonda, M.; Kauch, Anna; Janiš, Václav
2017-01-01
Roč. 131, č. 4 (2017), s. 1042-1044 ISSN 0587-4246 R&D Projects: GA ČR GA15-14259S Institutional support: RVO:68378271 Keywords : And erson model * parquet equations * numerical renormalization group Subject RIV: BM - Solid Matter Physics ; Magnetism OBOR OECD: Condensed matter physics (including formerly solid state physics, supercond.) Impact factor: 0.469, year: 2016
Czech Academy of Sciences Publication Activity Database
Feireisl, Eduard; Hošek, Radim; Maltese, D.; Novotný, A.
2017-01-01
Roč. 33, č. 4 (2017), s. 1208-1223 ISSN 0749-159X EU Projects: European Commission(XE) 320078 - MATH EF Institutional support: RVO:67985840 Keywords : convergence * error estimates * mixed numerical method * Navier–Stokes system Subject RIV: BA - General Math ematics OBOR OECD: Pure math ematics Impact factor: 1.079, year: 2016 http://onlinelibrary.wiley.com/doi/10.1002/num.22140/abstract
Numerical solution of the potential problem by integral equations without Green's functions
International Nuclear Information System (INIS)
De Mey, G.
1977-01-01
An integral equation technique will be presented to solve Laplace's equation in a two-dimensional area S. The Green's function has been replaced by a particular solution of Laplace equation in order to establish the integral equation. It is shown that accurate results can be obtained provided the pivotal elimination method is used to solve the linear algebraic set
Analytical and numerical solutions of the Schrödinger–KdV equation
Indian Academy of Sciences (India)
The Schrödinger–KdV equation with power-law nonlinearity is studied in this paper. The solitary wave ansatz method is used to carry out the integration of the equation and obtain one-soliton solution. The ′/ method is also used to integrate this equation. Subsequently, the variational iteration method and homotopy ...
Kmonodium, a Program for the Numerical Solution of the One-Dimensional Schrodinger Equation
Angeli, Celestino; Borini, Stefano; Cimiraglia, Renzo
2005-01-01
A very simple strategy for the solution of the Schrodinger equation of a particle moving in one dimension subjected to a generic potential is presented. This strategy is implemented in a computer program called Kmonodium, which is free and distributed under the General Public License (GPL).
Li, Y.; Ma, X.; Su, N.
2013-12-01
The movement of water and solute into and through the vadose zone is, in essence, an issue of immiscible displacement in pore-space network of a soil. Therefore, multiphase flow and transport in porous media, referring to three medium: air, water, and the solute, pose one of the largest unresolved challenges for porous medium fluid seepage. However, this phenomenon has always been largely neglected. It is expected that a reliable analysis model of the multi-phase flow in soil can truly reflect the process of natural movement about the infiltration, which is impossible to be observed directly. In such cases, geophysical applications of the nuclear magnetic resonance (NMR) provides the opportunity to measure the water movements into soils directly over a large scale from tiny pore to regional scale, accordingly enable it available both on the laboratory and on the field. In addition, the NMR provides useful information about the pore space properties. In this study, we proposed both laboratory and field experiments to measure the multi-phase flow parameters, together with optimize the model in computer programming based on the fractional partial differential equations (fPDE). In addition, we establish, for the first time, an infiltration model including solute flowing with water, which has huge influence on agriculture and soil environment pollution. Afterwards, with data collected from experiments, we simulate the model and analyze the spatial variability of parameters. Simulations are also conducted according to the model to evaluate the effects of airflow on water infiltration and other effects such as solute and absorption. It has significant meaning to oxygen irrigation aiming to higher crop yield, and shed more light into the dam slope stability. In summary, our framework is a first-time model added in solute to have a mathematic analysis with the fPDE and more instructive to agriculture activities.
Mohebbi, Akbar
2018-02-01
In this paper we propose two fast and accurate numerical methods for the solution of multidimensional space fractional Ginzburg-Landau equation (FGLE). In the presented methods, to avoid solving a nonlinear system of algebraic equations and to increase the accuracy and efficiency of method, we split the complex problem into simpler sub-problems using the split-step idea. For a homogeneous FGLE, we propose a method which has fourth-order of accuracy in time component and spectral accuracy in space variable and for nonhomogeneous one, we introduce another scheme based on the Crank-Nicolson approach which has second-order of accuracy in time variable. Due to using the Fourier spectral method for fractional Laplacian operator, the resulting schemes are fully diagonal and easy to code. Numerical results are reported in terms of accuracy, computational order and CPU time to demonstrate the accuracy and efficiency of the proposed methods and to compare the results with the analytical solutions. The results show that the present methods are accurate and require low CPU time. It is illustrated that the numerical results are in good agreement with the theoretical ones.
Solution of stochastic media transport problems using a numerical quadrature-based method
International Nuclear Information System (INIS)
Pautz, S. D.; Franke, B. C.; Prinja, A. K.; Olson, A. J.
2013-01-01
We present a new conceptual framework for analyzing transport problems in random media. We decompose such problems into stratified subproblems according to the number of material pseudo-interfaces within realizations. For a given subproblem we assign pseudo-interface locations in each realization according to product quadrature rules, which allows us to deterministically generate a fixed number of realizations. Quadrature integration of the solutions of these realizations thus approximately solves each subproblem; the weighted superposition of solutions of the subproblems approximately solves the general stochastic media transport problem. We revisit some benchmark problems to determine the accuracy and efficiency of this approach in comparison to randomly generated realizations. We find that this method is very accurate and fast when the number of pseudo-interfaces in a problem is generally low, but that these advantages quickly degrade as the number of pseudo-interfaces increases. (authors)
Hanson, R. K.; Presley, L. L.; Williams, E. V.
1972-01-01
The method of characteristics for a chemically reacting gas is used in the construction of the time-dependent, one-dimensional flow field resulting from the normal reflection of an incident shock wave at the end wall of a shock tube. Nonequilibrium chemical reactions are allowed behind both the incident and reflected shock waves. All the solutions are evaluated for oxygen, but the results are generally representative of any inviscid, nonconducting, and nonradiating diatomic gas. The solutions clearly show that: (1) both the incident- and reflected-shock chemical relaxation times are important in governing the time to attain steady state thermodynamic properties; and (2) adjacent to the end wall, an excess-entropy layer develops wherein the steady state values of all the thermodynamic variables except pressure differ significantly from their corresponding Rankine-Hugoniot equilibrium values.
Numerical solution of system of boundary value problems using B-spline with free parameter
Gupta, Yogesh
2017-01-01
This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation.
The NO-pair equation---Fundamental problems, numerical solutions and applications
International Nuclear Information System (INIS)
Martensson-Pendrill, A.
1989-01-01
The solution of inhomogeneous two-particle differential equations is a powerful method for treating correlation effects in the non-relativistic case and is reviewed briefly. The relativistic generalization, the so-called Dirac-Coulomb equation, is complicated by the need to distinguish between positive and negative energy states. The construction and use of projection operators onto positive energy states are presented and the application of the pair equation to other properties such as parity non-conservation is discussed
A comparison of numerical methods for the solution of continuous-time DSGE models
DEFF Research Database (Denmark)
Parra-Alvarez, Juan Carlos
This paper evaluates the accuracy of a set of techniques that approximate the solution of continuous-time DSGE models. Using the neoclassical growth model I compare linear-quadratic, perturbation and projection methods. All techniques are applied to the HJB equation and the optimality conditions...... parameters of the model and suggest the use of projection methods when a high degree of accuracy is required....
Numerical solution of continuous-time DSGE models under Poisson uncertainty
DEFF Research Database (Denmark)
Posch, Olaf; Trimborn, Timo
We propose a simple and powerful method for determining the transition process in continuous-time DSGE models under Poisson uncertainty numerically. The idea is to transform the system of stochastic differential equations into a system of functional differential equations of the retarded type. We...... classes of models. We illustrate the algorithm simulating both the stochastic neoclassical growth model and the Lucas model under Poisson uncertainty which is motivated by the Barro-Rietz rare disaster hypothesis. We find that, even for non-linear policy functions, the maximum (absolute) error is very...
Directory of Open Access Journals (Sweden)
Liquan Mei
2014-01-01
Full Text Available A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method.
Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator
Vabishchevich, P. N.
2018-03-01
A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order elliptic operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented.
Numerical solution of the controlled Duffing oscillator by semi-orthogonal spline wavelets
International Nuclear Information System (INIS)
Lakestani, M; Razzaghi, M; Dehghan, M
2006-01-01
This paper presents a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon compactly supported linear semi-orthogonal B-spline wavelets. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique
Numerical simulations of groundwater flow and solute transport in the Lake 233 aquifer
Energy Technology Data Exchange (ETDEWEB)
Klukas, M H; Moltyaner, G L
1995-05-01
A three-dimensional numerical flow model of the Lake 233 aquifer underlying the site of the proposed Intrusion Resistant Underground Structure (IRUS) for low level waste disposal is developed. A reference hydraulic conductivity distribution incorporating the key stratigraphic units and field estimates of recharge from Lake 233 are used as model input. The model was calibrated against the measured hydraulic head distribution, the flowpath of a historic {sup 90}Sr plume in the aquifer and measured groundwater velocities. (author). 23 refs., 4 tabs., 31 figs.
International Nuclear Information System (INIS)
Tashakor, S.; Jahanfarnia, G.; Hashemi-Tilehnoee, M.
2010-01-01
Point reactor kinetics equations are solved numerically using one group of delayed neutrons and with fuel burn-up and temperature feedback included. To calculate the fraction of one-group delayed neutrons, a group of differential equations are solved by an implicit time method. Using point reactor kinetics equations, changes in mean neutrons density, temperature, and reactivity are calculated in different times during the reactor operation. The variation of reactivity, temperature, and maximum power with time are compared with the predictions by other methods.
Exponential Convergence for Numerical Solution of Integral Equations Using Radial Basis Functions
Directory of Open Access Journals (Sweden)
Zakieh Avazzadeh
2014-01-01
Full Text Available We solve some different type of Urysohn integral equations by using the radial basis functions. These types include the linear and nonlinear Fredholm, Volterra, and mixed Volterra-Fredholm integral equations. Our main aim is to investigate the rate of convergence to solve these equations using the radial basis functions which have normic structure that utilize approximation in higher dimensions. Of course, the use of this method often leads to ill-posed systems. Thus we propose an algorithm to improve the results. Numerical results show that this method leads to the exponential convergence for solving integral equations as it was already confirmed for partial and ordinary differential equations.
Solution of the main problem of the lunar physical libration by a numerical method
Zagidullin, Arthur; Petrova, Natalia; Nefediev, Yurii
2016-10-01
Series of the lunar programs requires highly accurate ephemeris of the Moon at any given time. In the light of the new requirements on the accuracy the requirements to the lunar physical libration theory increase.At the Kazan University there is the experience of constructing the lunar rotation theory in the analytical approach. Analytical theory is very informative in terms of the interpretation of the observed data, but inferior to the accuracy of numerical theories. The most accurate numerical ephemeris of the Moon is by far the ephemeris DE430 / 431 built in the USA. It takes into account a large number of subtle effects both in external perturbations of the Moon, and in its internal structure. Before the Russian scientists the task is to create its own numerical theory that would be consistent with the American ephemeris. On the other hand, even the practical application of the american ephemeris requires a deep understanding of the principles of their construction and the intelligent application.As the first step, we constructed a theory in the framework of the main problem. Because we compare our theory with the analytical theory of Petrova (1996), all the constants and the theory of orbital motion are taken identical to the analytical theory. The maximum precision, which the model can provide is 0.01 seconds of arc, which is insufficient to meet the accuracy of modern observations, but this model provides the necessary basis for further development.We have constructed the system of the libration equations, for which the numerical integrator was developed. The internal accuracy of the software integrator is several nanoseconds. When compared with the data of Petrova the differences of order of 1 second are observed at the resonant frequencies. The reason, we believe, in the inaccuracy of the analytical theory. We carried out a comparison with the Eroshkin's data [2], which gave satisfactory agreement, and with Rambaux data. In the latter case, as expected
Numerical solutions of a ODE's system for neutronics
International Nuclear Information System (INIS)
Lima, Suzylaine da Silva; Ramos, Alexandre F.
2017-01-01
The preliminary results that were obtained in the computational implementation to solve numerically a System of Coupled Differential Equations were presented. This system is intended to describe the kinetics of nuclear reactions occurring in the interior of a fusion-fission hybrid reactor in which fusion occurs in periodic pulses, which may be laser, for example. The hybrid reactor contains a core in which the nuclear fusion fuel is injected and is enveloped by two layers both composed of subcritical fission fuel. Our results show that a fusion-fission hybrid reactor composed of two layers of fission can maximize the energy utilization in this type of reactor
Numerical simulations of groundwater flow and solute transport in the Lake 233 aquifer
International Nuclear Information System (INIS)
Klukas, M.H.; Moltyaner, G.L.
1995-05-01
A three-dimensional numerical flow model of the Lake 233 aquifer underlying the site of the proposed Intrusion Resistant Underground Structure (IRUS) for low level waste disposal is developed. A reference hydraulic conductivity distribution incorporating the key stratigraphic units and field estimates of recharge from Lake 233 are used as model input. The model was calibrated against the measured hydraulic head distribution, the flowpath of a historic 90 Sr plume in the aquifer and measured groundwater velocities. (author). 23 refs., 4 tabs., 31 figs
Directory of Open Access Journals (Sweden)
Reza Jalilian
2014-07-01
Full Text Available A Class of new methods based on a septic non-polynomial splinefunction for the numerical solution one-dimensional Bratu's problemare presented. The local truncation errors and the methods of order2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse ofsome band matrixes are obtained which are required in provingthe convergence analysis of the presented method. Associatedboundary formulas are developed. Convergence analysis of thesemethods is discussed. Numerical results are given to illustrate theefficiency of methods.
An Efficient and Robust Numerical Solution of the Full-Order Multiscale Model of Lithium-Ion Battery
Directory of Open Access Journals (Sweden)
Michal Beneš
2018-01-01
Full Text Available We propose a novel and efficient numerical approach for solving the pseudo two-dimensional multiscale model of the Li-ion cell dynamics based on first principles, describing the ion diffusion through the electrolyte and the porous electrodes, electric potential distribution, and Butler-Volmer kinetics. The numerical solution is obtained by the finite difference discretization of the diffusion equations combined with an original iterative scheme for solving the integral formulation of the laws of electrochemical interactions. We demonstrate that our implementation is fast and stable over the expected lifetime of the cell. In contrast to some simplified models, it provides physically consistent results for a wide range of applied currents including high loads. The algorithm forms a solid basis for simulations of cells and battery packs in hybrid electric vehicles, with possible straightforward extensions by aging and heat effects.
Yang, Jaw-Yen; Yan, Chih-Yuan; Diaz, Manuel; Huang, Juan-Chen; Li, Zhihui; Zhang, Hanxin
2014-01-08
The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al . 2012 Proc. R. Soc. A 468 , 1799-1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi-Dirac or Bose-Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas.
Numerical solution of the Navier--Stokes equations at high Reynolds numbers
International Nuclear Information System (INIS)
Shestakov, A.I.
1974-01-01
A numerical method is presented which is designed to solve the Navier-Stokes equations for two-dimensional, incompressible flow. The method is intended for use on problems with high Reynolds numbers for which calculations via finite difference methods have been unattainable or unreliable. The proposed scheme is a hybrid utilizing a time-splitting finite difference method in areas away from the boundaries. In areas neighboring the boundaries, the equations of motion are solved by the newly proposed vortex method by Chorin. The major accomplishment of the new scheme is that it contains a simple way for merging the two methods at the interface of the two subdomains. The proposed algorithm is designed for use on the time-dependent equations but can be used on steady state problems as well. The method is tested on the popular, time-independent, square cavity problem, an example of a separated flow with closed streamlines. Numerical results are presented for a Reynolds number of 10 3 . (auth)
Yang, Jaw-Yen; Yan, Chih-Yuan; Diaz, Manuel; Huang, Juan-Chen; Li, Zhihui; Zhang, Hanxin
2014-01-01
The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al. 2012 Proc. R. Soc. A 468, 1799–1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi–Dirac or Bose–Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas. PMID:24399919
Henclik, Sławomir
2018-03-01
The influence of dynamic fluid-structure interaction (FSI) onto the course of water hammer (WH) can be significant in non-rigid pipeline systems. The essence of this effect is the dynamic transfer of liquid energy to the pipeline structure and back, which is important for elastic structures and can be negligible for rigid ones. In the paper a special model of such behavior is analyzed. A straight pipeline with a steady flow, fixed to the floor with several rigid supports is assumed. The transient is generated by a quickly closed valve installed at the end of the pipeline. FSI effects are assumed to be present mainly at the valve which is fixed with a spring dash-pot attachment. Analysis of WH runs, especially transient pressure changes, for various stiffness and damping parameters of the spring dash-pot valve attachment is presented in the paper. The solutions are found analytically and numerically. Numerical results have been computed with the use of an own computer program developed on the basis of the four equation model of WH-FSI and the specific boundary conditions formulated at the valve. Analytical solutions have been found with the separation of variables method for slightly simplified assumptions. Damping at the dash-pot is taken into account within the numerical study. The influence of valve attachment parameters onto the WH courses was discovered and it was found the transient amplitudes can be reduced. Such a system, elastically attached shut-off valve in a pipeline or other, equivalent design can be a real solution applicable in practice.
International Nuclear Information System (INIS)
Kimura, Y.; Tokuyama, M.
2016-01-01
The full numerical solutions of the time-convolutionless modecoupling theory (TMCT) equation recently proposed by Tokuyama are compared with those of the ideal mode-coupling theory (MCT) equation based on the Percus- Yevick static structure factor for hard spheres qualitatively and quantitatively. The ergodic to non-ergodic transition at the critical volume fraction φ_c predicted by MCT is also shown to occur even for TMCT. Thus, φ_c of TMCT is shown to be much higher than that of MCT. The dynamics of coherent-intermediate scattering functions and their two-step relaxation process in a β stage are also discussed.
International Nuclear Information System (INIS)
Cunha Furtado, F. da; Galeao, A.C.N.R.
1984-01-01
A numerical procedure for the integration of the incompressible Navier-Stokes equations, when expressed in terms of a stream function equation and a vorticity transport equation, is presented. This procedure comprises: the variational formulation of the equations, the construction of the approximation spaces by the finite element method and the discretization via the Galerkin method. For the stationary problems, the system of non-linear algebraic equations resulting from the discretization is solved by the Newton-Raphson algorithm. Finally, for the transient problems, the solution of the non-linear ordinary differential equations resulting from the spatial discretization is accomplished through a Crank-Nicolson scheme. (Author) [pt
Antar, B. N.
1976-01-01
A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.
International Nuclear Information System (INIS)
Latynin, V.A.; Reshetov, V.A.; Karaseva, L.N.
1988-01-01
Numerical solution of the Stephen problem by the strained coordinate method is presented for an one-dimensional sphere. Differential formulae of heat fluxes from moving interfaces do not take into account volume heat capacities of the front nodes. Calculations, carried out according to these balanced formulae, as well as according to those usually used, have shown that the balanced formulae permit to reduce approximately by an order the number of nodes on the sphere radius, if similar accuracy of heat balance of the whole process of melting or crystallization is observed. 2 refs.; 1 fig
Gunzburger, M. D.; Nicolaides, R. A.
1986-01-01
Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.
The numerical analysis of eigenvalue problem solutions in the multigroup diffusion theory
International Nuclear Information System (INIS)
Woznick, Z.I.
1994-01-01
In this paper a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations is described. Usually the solution method is based on the system of inner and outer iterations. The presented matrix formalism allows us to visualize clearly, how the used matrix splitting influences the structure of the matrix in an eigenvalue problem to be solved as well as the independence between inner and outer iterations within global iterations. To keep the page limit, the present version of the paper consists only with first three of five sections given in the original paper under the same title (which will be published soon). (author). 13 refs
Directory of Open Access Journals (Sweden)
Naumenko Mikhail
2018-01-01
Full Text Available Modern parallel computing algorithm has been applied to the solution of the few-body problem. The approach is based on Feynman’s continual integrals method implemented in C++ programming language using NVIDIA CUDA technology. A wide range of 3-body and 4-body bound systems has been considered including nuclei described as consisting of protons and neutrons (e.g., 3,4He and nuclei described as consisting of clusters and nucleons (e.g., 6He. The correctness of the results was checked by the comparison with the exactly solvable 4-body oscillatory system and experimental data.
Naumenko, Mikhail; Samarin, Viacheslav
2018-02-01
Modern parallel computing algorithm has been applied to the solution of the few-body problem. The approach is based on Feynman's continual integrals method implemented in C++ programming language using NVIDIA CUDA technology. A wide range of 3-body and 4-body bound systems has been considered including nuclei described as consisting of protons and neutrons (e.g., 3,4He) and nuclei described as consisting of clusters and nucleons (e.g., 6He). The correctness of the results was checked by the comparison with the exactly solvable 4-body oscillatory system and experimental data.
Numerical Solution of Fractional Neutron Point Kinetics Model in Nuclear Reactor
Directory of Open Access Journals (Sweden)
Nowak Tomasz Karol
2014-06-01
Full Text Available This paper presents results concerning solutions of the fractional neutron point kinetics model for a nuclear reactor. Proposed model consists of a bilinear system of fractional and ordinary differential equations. Three methods to solve the model are presented and compared. The first one entails application of discrete Grünwald-Letnikov definition of the fractional derivative in the model. Second involves building an analog scheme in the FOMCON Toolbox in MATLAB environment. Third is the method proposed by Edwards. The impact of selected parameters on the model’s response was examined. The results for typical input were discussed and compared.
Solution of Euler unsteady equations using a second order numerical scheme
International Nuclear Information System (INIS)
Devos, J.P.
1992-08-01
In thermal power plants, the steam circuits experience incidents due to the noise and vibration induced by trans-sonic flow. In these configurations, the compressible fluid can be considered the perfect ideal. Euler equations therefore constitute a good model. However, processing of the discontinuities induced by the shockwaves are a particular problem. We give a bibliographical synthesis of the work done on this subject. The research by Roe and Harten leads to TVD (Total Variation Decreasing) type schemes. These second order schemes generate no oscillation and converge towards physically acceptable weak solutions. (author). 12 refs
Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method
Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad
2018-03-01
An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.
Numerical solution of problems concerning the thermal convection of a variable-viscosity liquid
Zherebiatev, I. F.; Lukianov, A. T.; Podkopaev, Iu. L.
A stabilizing-correction scheme is constructed for integrating the fourth-order equation describing the dynamics of a viscous incompressible liquid. As an example, a solution is obtained to the problem of the solidification of a liquid in a rectangular region with allowance for convective energy transfer in the liquid phase as well as temperature-dependent changes of viscosity. It is noted that the proposed method can be used to study steady-state problems of thermal convection in ingots obtained through continuous casting.
Salama, Amgad; Sun, Shuyu; Amin, Mohamed F. El
2015-01-01
In this work, the experimenting fields approach is applied to the numerical solution of the Navier-Stokes equation for incompressible viscous flow. In this work, the solution is sought for both the pressure and velocity fields in the same time. Apparently, the correct velocity and pressure fields satisfy the governing equations and the boundary conditions. In this technique a set of predefined fields are introduced to the governing equations and the residues are calculated. The flow according to these fields will not satisfy the governing equations and the boundary conditions. However, the residues are used to construct the matrix of coefficients. Although, in this setup it seems trivial constructing the global matrix of coefficients, in other setups it can be quite involved. This technique separates the solver routine from the physics routines and therefore makes easy the coding and debugging procedures. We compare with few examples that demonstrate the capability of this technique.
Energy Technology Data Exchange (ETDEWEB)
Jo, Jong Chull; Shin, Won Ky [Korea Institute of Nuclear Safety, Taejon (Korea, Republic of)
1997-12-31
This paper presents an effective and simple procedure for the simulation of the motion of the solid-liquid interfacial boundary and the transient temperature field during phase change process. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing the dual reciprocity boundary element method. The dual reciprocity boundary element approach provided in this paper is much simpler than the usual boundary element method applying a reciprocity principle and an available technique for dealing with domain integral of boundary element formulation simultaneously. The effectiveness of the present analysis method have been illustrated through comparisons of the calculation results of an example with its semi-analytical or other numerical solutions where available. 22 refs., 3 figs. (Author)
International Nuclear Information System (INIS)
Jo, Jong Chull; Shin, Won Ky
1997-01-01
This paper presents an effective and simple procedure for the simulation of the motion of the solid-liquid interfacial boundary and the transient temperature field during phase change process. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing the dual reciprocity boundary element method. The dual reciprocity boundary element approach provided in this paper is much simpler than the usual boundary element method applying a reciprocity principle and an available technique for dealing with domain integral of boundary element formulation simultaneously. The effectiveness of the present analysis method have been illustrated through comparisons of the calculation results of an example with its semi-analytical or other numerical solutions where available
Energy Technology Data Exchange (ETDEWEB)
Jo, Jong Chull; Shin, Won Ky [Korea Institute of Nuclear Safety, Taejon (Korea, Republic of)
1998-12-31
This paper presents an effective and simple procedure for the simulation of the motion of the solid-liquid interfacial boundary and the transient temperature field during phase change process. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing the dual reciprocity boundary element method. The dual reciprocity boundary element approach provided in this paper is much simpler than the usual boundary element method applying a reciprocity principle and an available technique for dealing with domain integral of boundary element formulation simultaneously. The effectiveness of the present analysis method have been illustrated through comparisons of the calculation results of an example with its semi-analytical or other numerical solutions where available. 22 refs., 3 figs. (Author)
Salama, Amgad
2015-06-01
In this work, the experimenting fields approach is applied to the numerical solution of the Navier-Stokes equation for incompressible viscous flow. In this work, the solution is sought for both the pressure and velocity fields in the same time. Apparently, the correct velocity and pressure fields satisfy the governing equations and the boundary conditions. In this technique a set of predefined fields are introduced to the governing equations and the residues are calculated. The flow according to these fields will not satisfy the governing equations and the boundary conditions. However, the residues are used to construct the matrix of coefficients. Although, in this setup it seems trivial constructing the global matrix of coefficients, in other setups it can be quite involved. This technique separates the solver routine from the physics routines and therefore makes easy the coding and debugging procedures. We compare with few examples that demonstrate the capability of this technique.
Numerical solutions to the critical state in a magnet-high temperature superconductor interaction
Energy Technology Data Exchange (ETDEWEB)
Ruiz-Alonso, D; Coombs, T A; Campbell, A M [Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ (United Kingdom)
2005-02-01
This paper presents an algorithm to simulate the electromagnetic behaviour of devices containing high temperature superconductors in axially symmetric problems. The numerical method is built within the finite element method. The electromagnetic properties of HTSCs are described through the critical-state model. Measurements of the axial force between a permanent magnet and a melt-textured YBCO puck are obtained in order to validate the method. This simple system is modelled so that the proposed method obtains the current distribution and electromagnetic fields in the HTSC. The forces in the interaction between the magnet and the HTSC puck can then be calculated. A comparison between experimental and simulation results shows good matching. The simplification of using the critical-state model and ignoring flux creep in this type of configuration is also explored.
Numerical solution of electromagnetic field problems in two and three dimensions
International Nuclear Information System (INIS)
Trowbridge, C.W.
1981-01-01
Recent developments in algorithms for solving electromagnetic field problems carried out at Rutherford Appleton Laboratory (RAL) are reviewed. The interaction of electric and magnetic fields provides many examples of coupled problems which have been solved by the Finite Element method. This paper concentrates on static and low frequency problems using the differential operator approach. The status of computation for 2D fields is discussed. The use of scalar potentials for 3D static fields for economy is emphasised and the importance of selecting potential types carefully to minimise numerical cancellation errors is also discussed. Some formulations for the vector 3D field problem for eddy current fields are derived with analytic and experimental field measurement comparisons. Results using software packages built at RAL are presented to illustrate the methods. (author)
Numerical solution of the model problem of CCRF-discharge at atmospheric pressure
Directory of Open Access Journals (Sweden)
Zheltukhin Viktor
2017-01-01
Full Text Available This work describes a 1D mathematical model of capacitive coupled RF discharge between symmetrical electrodes in argon at atmospheric pressure in a local approximation. Electrons, atomic and molecular ions, metastable atoms and argon dimmers as well as ground-state atoms are considered. A simplified diagram of argon excited states when two metastable and two resonance states are replaced with the uniform level. Such diagram is frequently used to simulate argon plasma due to efficient mixing of these layers at electron impacts. Velocity factors of electron impact processes were calculated using Boltzmann equation with a glance to electron-electron collisions. This work describes numerical algorithm of mathematical model implementation, which is based on finite-dimensional approximation of the problem using difference schemes together with iteration process. The software was developed to implement iterative processes using MatLab. Characteristics of atmospheric pressure capacitive coupled RF discharge at interelectrod distance 20 mm are calculated.
A numerical solution model of the rewetting of a nuclear fuel rod
International Nuclear Information System (INIS)
Braz Filho, F.A.
1984-01-01
The study of thermal behaviour of a nuclear reactor fuel rod during the reflooding phase of the loss-of-coolant accident (LOCA) is presented. A mathematical model and a numerical scheme were proposed in order to solve the bidimensional heat conduction equation in cylindrical coordinates. The phenomenon of reflooding is not completely understood. One of the main difficulties is to estimate the heat transfer coefficient (h). For this reason two different models were elaborated: in the first three regions are considered and in each region h is considered constant; in the second the h profile is adjusted according to the boiling curve. The three region model yields satisfactory results at high and low mass flows while the 'boiling curve' model yields reasonable at low flows. (Author) [pt
Numerical solutions of anharmonic vibration of BaO and SrO molecules
Energy Technology Data Exchange (ETDEWEB)
Pramudito, Sidikrubadi; Sanjaya, Nugraha Wanda [Theoretical Physics Division, Department of Physics, Bogor Agricultural University, Jalan Meranti Kampus IPB Dramaga Bogor 16680 (Indonesia); Sumaryada, Tony, E-mail: tsumaryada@ipb.ac.id [Theoretical Physics Division, Department of Physics, Bogor Agricultural University, Jalan Meranti Kampus IPB Dramaga Bogor 16680 (Indonesia); Computational Biophysics and Molecular Modeling Research Group (CBMoRG), Department of Physics, Bogor Agricultural University, Jalan Meranti Kampus IPB Dramaga Bogor 16680 (Indonesia)
2016-03-11
The Morse potential is a potential model that is used to describe the anharmonic behavior of molecular vibration between atoms. The BaO and SrO molecules, which are two almost similar diatomic molecules, were investigated in this research. Some of their properties like the value of the dissociation energy, the energy eigenvalues of each energy level, and the profile of the wavefunctions in their correspondence vibrational states were presented in this paper. Calculation of the energy eigenvalues and plotting the wave function’s profiles were performed using Numerov method combined with the shooting method. In general we concluded that the Morse potential solved with numerical methods could accurately produce the vibrational properties and the wavefunction behavior of BaO and SrO molecules from the ground state to the higher states close to the dissociation level.
Numerical solution of the Black-Scholes equation using cubic spline wavelets
Černá, Dana
2016-12-01
The Black-Scholes equation is used in financial mathematics for computation of market values of options at a given time. We use the θ-scheme for time discretization and an adaptive scheme based on wavelets for discretization on the given time level. Advantages of the proposed method are small number of degrees of freedom, high-order accuracy with respect to variables representing prices and relatively small number of iterations needed to resolve the problem with a desired accuracy. We use several cubic spline wavelet and multi-wavelet bases and discuss their advantages and disadvantages. We also compare an isotropic and anisotropic approach. Numerical experiments are presented for the two-dimensional Black-Scholes equation.
Zhao, Yu-long; Tang, Xu-chuan; Zhang, Lie-hui; Tang, Hong-ming; Tao, Zheng-Wu
2018-06-01
The multiscale pore size and specific gas storage mechanism in organic-rich shale gas reservoirs make gas transport in such reservoirs complicated. Therefore, a model that fully incorporates all transport mechanisms and employs an accurate numerical method is urgently needed to simulate the gas production process. In this paper, a unified model of apparent permeability was first developed, which took into account multiple influential factors including slip flow, Knudsen diffusion (KD), surface diffusion, effects of the adsorbed layer, permeability stress sensitivity, and ad-/desorption phenomena. Subsequently, a comprehensive mathematical model, which included the model of apparent permeability, was derived to describe gas production behaviors. Thereafter, on the basis of unstructured perpendicular bisection grids and finite volume method, a fully implicit numerical simulator was developed using Matlab software. The validation and application of the new model were confirmed using a field case reported in the literature. Finally, the impacts of related influencing factors on gas production were analyzed. The results showed that KD resulted in a negligible impact on gas production in the proposed model. The smaller the pore size was, the more obvious the effects of the adsorbed layer on the well production rate would be. Permeability stress sensitivity had a slight effect on well cumulative production in shale gas reservoirs. Adsorbed gas made a major contribution to the later flow period of the well; the greater the adsorbed gas content, the greater the well production rate would be. This paper can improve the understanding of gas production in shale gas reservoirs for petroleum engineers.
Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation
Poleshkin, Sergey O.; Malkov, Ewgenij A.; Kudryavtsev, Alexey N.; Shershnev, Anton A.; Bondar, Yevgeniy A.; Kohanchik, A. A.
2017-10-01
There are various methods for calculating rarefied gas flows, in particular, statistical methods and deterministic methods based on the finite-difference solutions of the Boltzmann nonlinear kinetic equation and on the solutions of model kinetic equations. There is no universal method; each has its disadvantages in terms of efficiency or accuracy. The choice of the method depends on the problem to be solved and on parameters of calculated flows. Qualitative theoretical arguments help to determine the range of parameters of effectively solved problems for each method; however, it is advisable to perform comparative tests of calculations of the classical problems performed by different methods and with different parameters to have quantitative confirmation of this reasoning. The paper provides the results of the calculations performed by the authors with the help of the Direct Simulation Monte Carlo method and finite-difference methods of solving the Boltzmann equation and model kinetic equations. Based on this comparison, conclusions are made on selecting a particular method for flow simulations in various ranges of flow parameters.
Directory of Open Access Journals (Sweden)
Suheel Abdullah Malik
Full Text Available In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE through substitution is converted into a nonlinear ordinary differential equation (NODE. The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM, homotopy perturbation method (HPM, and optimal homotopy asymptotic method (OHAM, show that the suggested scheme is fairly accurate and viable for solving such problems.
Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul
2015-01-01
In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.
International Nuclear Information System (INIS)
Suzuki, Shunichi; Motoshima, Takayuki; Naemura, Yumi; Kubo, Shin; Kanie, Shunji
2009-01-01
The authors develop a numerical code based on Local Discontinuous Galerkin Method for transient groundwater flow and reactive solute transport problems in order to make it possible to do three dimensional performance assessment on radioactive waste repositories at the earliest stage possible. Local discontinuous Galerkin Method is one of mixed finite element methods which are more accurate ones than standard finite element methods. In this paper, the developed numerical code is applied to several problems which are provided analytical solutions in order to examine its accuracy and flexibility. The results of the simulations show the new code gives highly accurate numeric solutions. (author)
Glowinski, R.; Dean, E.J.; Guidoboni, G.; Juárez, L.H.; Pan, T.-W.
2008-01-01
The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge--Ampère equation. The results of numerical ...
International Nuclear Information System (INIS)
Trowbridge, C.W.
1976-06-01
Various integral equation methods are described. For magnetostatic problems three formulations are considered in detail, (a) the direct solution method for the magnetisation distribution in permeable materials, (b) a method based on a scalar potential and (c) the use of an integral equation derived from Green's Theorem, i.e. the so-called Boundary Integral Method (BIM). In the case of (a) results are given for two-and three-dimensional non-linear problems with comparisons against measurement. For methods (b) and (c) which both lead to a more economic use of the computer than (a) some preliminary results are given for simple cases. For eddy current problems various methods are discussed and some results are given from a computer program based on a vector potential formulation. (author)
On solution of Maxwell's equations in axisymmetric domains with edges. Part II: Numerical aspects
International Nuclear Information System (INIS)
Nkemzi, Boniface
2003-10-01
In this paper we consider the Fourier-finite-element method for treating the Maxwell's equations in three-dimensional axisymmetric domains with reentrant edges. By means of partial Fourier analysis, the 3D BVP is decomposed into an infinite sequence of 2D variational equations in the plane meridian domain of the axisymmetric domain, a finite number of which is considered and treated using nodal H 1 -conforming finite elements. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions. Emphases are given to estimates of the Fourier-finite-element approximation error and convergence analysis in the H 1 -norm under different regularity assumptions. (author)
Energy Technology Data Exchange (ETDEWEB)
Trowbridge, C W
1976-06-01
Various integral equation methods are described. For magnetostatic problems three formulations are considered in detail, (a) the direct solution method for the magnetisation distribution in permeable materials, (b) a method based on a scalar potential, and (c) the use of an integral equation derived from Green's Theorem, i.e. the so-called Boundary Integral Method (BIM). In the case of (a) results are given for two-and three-dimensional non-linear problems with comparisons against measurement. For methods (b) and (c), which both lead to a more economical use of the computer than (a), some preliminary results are given for simple cases. For eddy current problems various methods are discussed and some results are given from a computer program based on a vector potential formulation.
International Nuclear Information System (INIS)
Sanchez G, J.
2007-01-01
A standard procedure for the solution of singular integral equations is applied to the one-dimensional transport equation for monoenergetic neutrons. The results obtained with two versions of the procedure, differing only in the extent of the basic region to which they are applied, are compared with analytically derived results available for benchmarking. The procedures considered yield consistent results for the calculated neutron densities and eigenvalues. Several approximate expressions of the neutron density are used to render closed-form formulas for the densities which can then be analytically operated on to obtain expressions for extrapolation distances or angular densities or serve other purposes that require an analytical expression of the neutron density. (Author)
Collier, Richard S.; McKenna, Paul M.; Perala, Rodney A.
1991-08-01
The objective here is to describe the lightning hazards to buildings and their internal environments using advanced formulations of Maxwell's Equations. The method described is the Three Dimensional Finite Difference Time Domain Solution. It can be used to solve for the lightning interaction with such structures in three dimensions with the inclusion of a considerable amount of detail. Special techniques were developed for including wire, plumbing, and rebar into the model. Some buildings have provisions for lightning protection in the form of air terminals connected to a ground counterpoise system. It is shown that fields and currents within these structures can be significantly high during a lightning strike. Time lapse video presentations were made showing the electric and magnetic field distributions on selected cross sections of the buildings during a simulated lightning strike.
Numerical solution for gate induced vibration due to under flow cavitation
International Nuclear Information System (INIS)
Sadrnezhad, S. A.
2001-01-01
Among the many forces to which hydraulic structures are exposed to, the forces induced by cavitation incident are of typical hydrodynamic unknown forces. The aim of this study is to define these forces as coupled fluid-structure interaction under two dynamic effects. The first dynamic effect which incorporates facilities for dealing with cavitation fluid is based on the appearance and bursting of vapor bubbles. The second hydrodynamic effect is dynamic excitation mechanism of the structure. In fluid-structure interaction, both the structure behavior and fluid are considered linear. Fluids can take some tension the extent of which depends on concentration and size of micro bubbles present; nevertheless, if the absolute pressure drops to a value close to the vapor pressure of the fluid, bubbles are formed and cavitation phenomena occurs. In this paper a fixed-wheel gate under the head pressure of a reservoir is considered to be affected by under flow cavitation. Normally, partially opened gates induce energy dissipation resulting in high turbulence, causing negative pressure and cavitation at the back and this exits the gate vibration. Moreover, there are several mechanisms which may cause heavy, self-excited vibration. According to the proposed method, a time function presenting the oscillation and pressure fluctuation in the vicinity of gate lip is estimated. This estimation is based on the parameters obtained from a two dimensional solution of flow under the gate lip. Accordingly, periodic time variable nodal forces are calculated and applied to gate lip element nodes. A transient dynamic solution of the gate, while its lip is sustaining nodal forces is estimated as time function. The results for the most server modal deformation of the structure time history of some critical elements and variation of equivalent force versus time are presented
Numerical solution of multiband k.p model for tunnelling in type-II heterostructures
Directory of Open Access Journals (Sweden)
A.E. Botha
2010-01-01
Full Text Available A new and very general method was developed for calculating the charge and spin-resolved electron tunnelling in type-II heterojunctions. Starting from a multiband k.p description of the bulk energy-band structure, a multiband k.p Riccati equation was derived. The reflection and transmission coefficients were obtained for each channel by integrating the Riccati equation over the entire heterostructure. Numerical instability was reduced through this method, in which the multichannel log-derivative of the envelope function matrix, rather than the envelope function itself, was propagated. As an example, a six-band k.p Hamiltonian was used to calculate the current-voltage characteristics of a 10-nm wide InAs/ GaSb/InAs single quantum well device which exhibited negative differential resistance at room temperature. The calculated current as a function of applied (bias voltage was found to be in semiquantitative agreement with the experiment, a result which indicated that inelastic transport mechanisms do not contribute significantly to the valley currents measured in this particular device.
Computational time analysis of the numerical solution of 3D electrostatic Poisson's equation
Kamboh, Shakeel Ahmed; Labadin, Jane; Rigit, Andrew Ragai Henri; Ling, Tech Chaw; Amur, Khuda Bux; Chaudhary, Muhammad Tayyab
2015-05-01
3D Poisson's equation is solved numerically to simulate the electric potential in a prototype design of electrohydrodynamic (EHD) ion-drag micropump. Finite difference method (FDM) is employed to discretize the governing equation. The system of linear equations resulting from FDM is solved iteratively by using the sequential Jacobi (SJ) and sequential Gauss-Seidel (SGS) methods, simulation results are also compared to examine the difference between the results. The main objective was to analyze the computational time required by both the methods with respect to different grid sizes and parallelize the Jacobi method to reduce the computational time. In common, the SGS method is faster than the SJ method but the data parallelism of Jacobi method may produce good speedup over SGS method. In this study, the feasibility of using parallel Jacobi (PJ) method is attempted in relation to SGS method. MATLAB Parallel/Distributed computing environment is used and a parallel code for SJ method is implemented. It was found that for small grid size the SGS method remains dominant over SJ method and PJ method while for large grid size both the sequential methods may take nearly too much processing time to converge. Yet, the PJ method reduces computational time to some extent for large grid sizes.
Dehghan, Mehdi; Nikpour, Ahmad
2013-09-01
In this research, we propose two different methods to solve the coupled Klein-Gordon-Zakharov (KGZ) equations: the Differential Quadrature (DQ) and Globally Radial Basis Functions (GRBFs) methods. In the DQ method, the derivative value of a function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The principal work in this method is the determination of weight coefficients. We use two ways for obtaining these coefficients: cosine expansion (CDQ) and radial basis functions (RBFs-DQ), the former is a mesh-based method and the latter categorizes in the set of meshless methods. Unlike the DQ method, the GRBF method directly substitutes the expression of the function approximation by RBFs into the partial differential equation. The main problem in the GRBFs method is ill-conditioning of the interpolation matrix. Avoiding this problem, we study the bases introduced in Pazouki and Schaback (2011) [44]. Some examples are presented to compare the accuracy and easy implementation of the proposed methods. In numerical examples, we concentrate on Inverse Multiquadric (IMQ) and second-order Thin Plate Spline (TPS) radial basis functions. The variable shape parameter (exponentially and random) strategies are applied in the IMQ function and the results are compared with the constant shape parameter.
Ab initio theory of superconductivity in a magnetic field. II. Numerical solution
Linscheid, A.; Sanna, A.; Gross, E. K. U.
2015-07-01
We numerically investigate the spin density functional theory for superconductors (SpinSCDFT) and the approximated exchange-correlation functional, derived and presented in the preceding Paper I [A. Linscheid et al., Phys. Rev. B 92, 024505 (2015), 10.1103/PhysRevB.92.024505]. As a test system, we employ a free-electron gas featuring an exchange splitting, a phononic pairing field, and a Coulomb repulsion. SpinSCDFT results are compared with Sarma, the Bardeen-Cooper-Schrieffer theory, and with an Eliashberg type of approach. We find that the spectrum of the superconducting Kohn-Sham SpinSCDFT system is not in agreement with the true quasiparticle structure. Therefore, starting from the Dyson equation, we derive a scheme that allows to compute the many-body excitations of the superconductor and represents the extension to superconductivity of the G0W0 method in band-structure theory. This superconducting G0W0 method vastly improves the predicted spectra.
Numerical solution of the 1D kinetics equations using a cubic reduced nodal scheme
International Nuclear Information System (INIS)
Gomez T, A.M.; Valle G, E. del; Delfin L, A.; Alonso V, G.
2003-01-01
In this work a finite differences technique centered in mesh based on a cubic reduced nodal scheme type finite element to solve the equations of the kinetics 1 D that include the equations corresponding to the concentrations of precursors of delayed neutrons is described. The technique of finite elements used is that of Galerkin where so much the neutron flux as the concentrations of precursors its are spatially approached by means of a three grade polynomial. The matrices of rigidity and of mass that arise during this discretization process are numerically evaluated using the open quadrature non standard of Newton-Cotes and that of Radau respectively. The purpose of the application of these quadratures is the one of to eliminate in the global matrices the couplings among the values of the flow in points of the discretization with the consequent advantages as for the reduction of the order of the matrix associated to the discreet problem that is to solve. As for the time dependent part the classical integration scheme known as Θ scheme is applied. After carrying out the one reordering of unknown and equations it arrives to a reduced system that it can be solved but quickly. With the McKin compute program developed its were solved three benchmark problems and those results are shown for the relative powers. (Author)
Energy Technology Data Exchange (ETDEWEB)
Kotler, Z.; Neria, E.; Nitzan, A. (Tel Aviv Univ. (Israel). School of Chemistry)
1991-02-01
The use of the time-dependent self-consistent field approximation (TDSCF) in the numerical solution of quantum curve crossing and tunneling dynamical problems is investigated. Particular emphasis is given to multiconfiguration TDSCF (MCTDSCF) approximations, which are shown to perform considerably better with only a small increase in computational effort. We investigate a number of simple models in which a 'system' characterized by two electronic potential surfaces evolves while interacting with a 'bath' mode described by an harmonic oscillator, and compare exact numerical solutions to one- and two-configuration TDSCF approximations. We also introduce and investigate a semiclassical approximation in which the 'bath' mode is described by semiclassical wavepackets (one for each electronic state) and show that for all models investigated this scheme works very well in comparison with the fully quantum MCTDSCF approximation. This provides a potentially very useful method to simulate strongly quantum systems coupled to an essentially classical environment. (orig.).
Amarti, Z.; Nurkholipah, N. S.; Anggriani, N.; Supriatna, A. K.
2018-03-01
Predicting the future of population number is among the important factors that affect the consideration in preparing a good management for the population. This has been done by various known method, one among them is by developing a mathematical model describing the growth of the population. The model usually takes form in a differential equation or a system of differential equations, depending on the complexity of the underlying properties of the population. The most widely used growth models currently are those having a sigmoid solution of time series, including the Verhulst logistic equation and the Gompertz equation. In this paper we consider the Allee effect of the Verhulst’s logistic population model. The Allee effect is a phenomenon in biology showing a high correlation between population size or density and the mean individual fitness of the population. The method used to derive the solution is the Runge-Kutta numerical scheme, since it is in general regarded as one among the good numerical scheme which is relatively easy to implement. Further exploration is done via the fuzzy theoretical approach to accommodate the impreciseness of the initial values and parameters in the model.
International Nuclear Information System (INIS)
Inc, Mustafa
2007-01-01
In this Letter, a scheme is developed to study numerical doubly-periodic solutions of the (2+1)-dimensional Boussinesq equation with initial condition by the variational iteration method. As a result, the approximate and exact doubly-periodic solutions are obtained. For different modulus m, comparison between the approximate solution and the exact solution is made graphically, revealing that the variational iteration method is a powerful and effective tool to non-linear problems
International Nuclear Information System (INIS)
Gustafsson, Lars-Goeran; Sassner, Mona; Bosson, Emma
2008-12-01
The Swedish Nuclear Fuel and Waste Management Company (SKB) is performing site investigations at two different locations in Sweden, referred to as the Forsmark and Laxemar areas, with the objective of siting a final repository for high-level radioactive waste. Data from the site investigations are used in a variety of modelling activities. This report presents model development and results of numerical transport modelling based on the numerical flow modelling of surface water and near-surface groundwater at the Forsmark site. The numerical modelling was performed using the modelling tool MIKE SHE and is based on the site data and conceptual model of the Forsmark areas. This report presents solute transport applications based on both particle tracking simulations and advection-dispersion calculations. The MIKE SHE model is the basis for the transport modelling presented in this report. Simulation cases relevant for the transport from a deep geological repository have been studied, but also the pattern of near surface recharge and discharge areas. When the main part of the modelling work presented in this report was carried out, the flow modelling of the Forsmark site was not finalised. Thus, the focus of this work is to describe the sensitivity to different transport parameters, and not to point out specific areas as discharge areas from a future repository (this is to be done later, within the framework of the safety assessment). In the last chapter, however, results based on simulations with the re-calibrated MIKE SHE flow model are presented. The results from the MIKE SHE water movement calculations were used by cycling the calculated transient flow field for a selected one-year period as many times as needed to achieve the desired simulation period. The solute source was located either in the bedrock or on top of the model. In total, 15 different transport simulation cases were studied. Five of the simulations were particle tracking simulations, whereas the rest
Directory of Open Access Journals (Sweden)
S. A. Eftekhari
Full Text Available AbstractThe differential quadrature method (DQM is one of the most elegant and efficient methods for the numerical solution of partial differential equations arising in engineering and applied sciences. It is simple to use and also straightforward to implement. However, the DQM is well-known to have some difficulty when applied to partial differential equations involving singular functions like the Dirac-delta function. This is caused by the fact that the Dirac-delta function cannot be directly discretized by the DQM. To overcome this difficulty, this paper presents a simple differential quadrature procedure in which the Dirac-delta function is replaced by regularized smooth functions. By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. To demonstrate the applicability and reliability of the proposed method, it is applied here to solve some moving load problems of beams and rectangular plates, where the location of the moving load is described by a time-dependent Dirac-delta function. The results generated by the proposed method are compared with analytical and numerical results available in the literature. Numerical results reveal that the proposed method can be used as an efficient tool for dynamic analysis of beam- and plate-type structures traversed by moving dynamic loads.
Directory of Open Access Journals (Sweden)
Jintao Song
2015-01-01
Full Text Available The foundation boundaries of numerical simulation models of hydraulic structures dominated by a vertical load are investigated. The method used is based on the stress formula for fundamental solutions to semi-infinite space body elastic mechanics under a vertical concentrated force. The limit method is introduced into the original formula, which is then partitioned and analyzed according to the direction of the depth extension of the foundation. The point load will be changed to a linear load with a length of 2a. Inverse proportion function assumptions are proposed at parameter a and depth l of the calculation points to solve the singularity questions of elastic stress in a semi-infinite space near the ground. Compared with the original formula, changing the point load to a linear load with a length of 2a is more reasonable. Finally, the boundary depth criterion of a hydraulic numerical simulation model is derived and applied to determine the depth boundary formula for gravity dam numerical simulations.
minatti, L.
2013-12-01
A finite volume model solving the shallow water equations coupled with the sediments continuity equation in composite channels with irregular geometry is presented. The model is essentially 1D but can handle composite cross-sections in which bedload transport is considered to occur inside the main channel only. This assumption is coherent with the observed behavior of rivers on short time scales where main channel areas exhibit more relevant morphological variations than overbanks. Furthermore, such a model allows a more precise prediction of thalweg elevation and cross section shape variations than fully 1D models where bedload transport is considered to occur uniformly over the entire cross section. The coupling of the equations describing water and sediments dynamics results in a hyperbolic non-conservative system that cannot be solved numerically with the use of a conservative scheme. Therefore, a path-conservative scheme, based on the approach proposed by Pares and Castro (2004) has been devised in order to account for the coupling with the sediments continuity equation and for the concurrent presence of bottom elevation and breadth variations of the cross section. In order to correctly compute numerical fluxes related to bedload transport in main channel areas, a special treatment of the equations is employed in the model. The resulting scheme is well balanced and fully coupled and can accurately model abrupt time variations of flow and bedload transport conditions in wide rivers, characterized by the presence of overbank areas that are less active than the main channel. The accuracy of the model has been first tested in fixed bed conditions by solving problems with a known analytical solution: in these tests the model proved to be able to handle shocks and supercritical flow conditions properly(see Fig. 01). A practical application of the model to the Ombrone river, southern Tuscany (Italy) is shown. The river has shown relevant morphological changes during
Directory of Open Access Journals (Sweden)
Basuki Widodo
2012-02-01
Full Text Available Corrosion process is a natural case that happened at the various metals, where the corrosion process in electrochemical can be explained by using galvanic cell. The iron corrosion process is based on the acidity degree (pH of a condensation, iron concentration and condensation temperature of electrolyte. Those are applied at electrochemistry cell. The iron corrosion process at this electrochemical cell also able to generate electrical potential and electric current during the process takes place. This paper considers how to build a mathematical model of iron corrosion, electrical potential and electric current. The mathematical model further is solved using the finite element method. This iron corrosion model is built based on the iron concentration, condensation temperature, and iteration time applied. In the electric current density model, the current based on electric current that is happened at cathode and anode pole and the iteration time applied. Whereas on the potential electric model, it is based on the beginning of electric potential and the iteration time applied. The numerical results show that the part of iron metal, that is gristle caused by corrosion, is the part of metal that has function as anode and it has some influences, such as time depth difference, iron concentration and condensation temperature on the iron corrosion process and the sum of reduced mass during corrosion process. Moreover, difference influence of time and beginning electric potential has an effect on the electric potential, which emerges during corrosion process at the electrochemical cell. Whereas, at the electrical current is also influenced by difference of depth time and condensation temperature applied.Keywords: Iron Corrosion, Concentration of iron, Electrochemical Cell and Finite Element Method
Some strange numerical solutions of the non-stationary Navier-Stokes equations in pipes
Energy Technology Data Exchange (ETDEWEB)
Rummler, B.
2001-07-01
A general class of boundary-pressure-driven flows of incompressible Newtonian fluids in three-dimensional pipes with known steady laminar realizations is investigated. Considering the laminar velocity as a 3D-vector-function of the cross-section-circle arguments, we fix the scale for the velocity by the L{sub 2}-norm of the laminar velocity. The usual new variables are introduced to get dimension-free Navier-Stokes equations. The characteristic physical and geometrical quantities are subsumed in the energetic Reynolds number Re and a parameter {psi}, which involves the energetic ratio and the directions of the boundary-driven part and the pressure-driven part of the laminar flow. The solution of non-stationary dimension-free Navier-Stokes equations is sought in the form u=u{sub L}+u, where u{sub L} is the scaled laminar velocity and periodical conditions in center-line-direction are prescribed for u. An autonomous system (S) of ordinary differential equations for the time-dependent coefficients of the spatial Stokes eigenfunction is got by application of the Galerkin-method to the dimension-free Navier-Stokes equations for u. The finite-dimensional approximations u{sub N({lambda}}{sub )} of u are defined in the usual way. (orig.)
Application of preconditioned GMRES to the numerical solution of the neutron transport equation
International Nuclear Information System (INIS)
Patton, B.W.; Holloway, J.P.
2002-01-01
The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU(τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU(τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU(τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements
International Nuclear Information System (INIS)
Monticelli, Cintia O.; Wortmann, Sergio; Segatto, Cynthia F.
2005-01-01
In this work is obtained a hybrid solution to the Fokker-Planck equation with energy dependency, very used in ion implantation problems. The main idea relies on the application of Laplace transform in the energy variable, and finite-difference in the spatial variable and in the angular variable. This procedure leads to a symbolic matrix problem for the transformed energy. To solve this system, is needed to do the Laplace inverse of the (sI+A) matrix, where s is a complex parameter, I is the identity matrix and A is a square matrix that was proceeded from the finite-difference in the spatial variable and in the angular variable. The matrix A is not defective, then is taken decomposition of A in a sum of two others matrices, where one is defective. It leads a iterative inversion method, similar the source fixed method combined with the diagonalization method, then is obtained the values to the angular flux. Hereafter we can to determine the energy deposited into the electronic system and in the nuclear system of the target. To comprove the results obtained, the simulation of implantation of B into Si at energies ranging from 1 KeV to 50 MeV was carried out and compared with the results by software SRIM2003. (author)
Energy Technology Data Exchange (ETDEWEB)
Gustafsson, Lars-Goeran; Sassner, Mona (DHI Sverige AB, Stockholm (Sweden)); Bosson, Emma (Swedish Nuclear Fuel and Waste Management Co., Stockholm (Sweden))
2008-12-15
The Swedish Nuclear Fuel and Waste Management Company (SKB) is performing site investigations at two different locations in Sweden, referred to as the Forsmark and Laxemar areas, with the objective of siting a final repository for high-level radioactive waste. Data from the site investigations are used in a variety of modelling activities. This report presents model development and results of numerical transport modelling based on the numerical flow modelling of surface water and near-surface groundwater at the Forsmark site. The numerical modelling was performed using the modelling tool MIKE SHE and is based on the site data and conceptual model of the Forsmark areas. This report presents solute transport applications based on both particle tracking simulations and advection-dispersion calculations. The MIKE SHE model is the basis for the transport modelling presented in this report. Simulation cases relevant for the transport from a deep geological repository have been studied, but also the pattern of near surface recharge and discharge areas. When the main part of the modelling work presented in this report was carried out, the flow modelling of the Forsmark site was not finalised. Thus, the focus of this work is to describe the sensitivity to different transport parameters, and not to point out specific areas as discharge areas from a future repository (this is to be done later, within the framework of the safety assessment). In the last chapter, however, results based on simulations with the re-calibrated MIKE SHE flow model are presented. The results from the MIKE SHE water movement calculations were used by cycling the calculated transient flow field for a selected one-year period as many times as needed to achieve the desired simulation period. The solute source was located either in the bedrock or on top of the model. In total, 15 different transport simulation cases were studied. Five of the simulations were particle tracking simulations, whereas the rest
Hage, Ilige S; Hamade, Ramsey F
2017-09-01
solution are corroborated experimentally using microhardness indentation measurements taken at the same points that the digital images were taken along a radial distance emanating from the interior (endosteum) surface toward the bone's exterior (periosteum) surface. Good agreement was found between numerically calculated and indentation measured stiffness of Intracortical lamellae. Both indentation measurements and numerical solutions of matrix stiffness showed increasing linear trend of compressive longitudinal modulus (E11) values vs. radial position for both interior and exterior regions. In the interior (exterior) region of cortical bone, stiffness modulus values were found to range from 18.5 to 23.4 GPa (23 to 26.0 GPa) with the aggregate stiffness of the cortical lamella in the exterior region being 12% stiffer than that in the interior region. In order to further validate these findings, experimental and FEM simulation of a mid-diaphysis bone ring under compression is employed. The FEM numerical deflections employed nine concentric regions across the thickness with graded stiffness values based on the digital segmentation and homogenization scheme. Bone ring deflections are found to agree well with measured deformations of the compression bone ring.
Abbott, B. P.; Abbott, R.; Abbott, T. D.; Abernathy, M. R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R. X.; Adya, V. B.; Affeldt, C.; Agathos, M.; Agatsuma, K.; Aggarwal, N.; Aguiar, O. D.; Aiello, L.; Ain, A.; Ajith, P.; Allen, B.; Allocca, A.; Altin, P. A.; Anderson, S. B.; Anderson, W. G.; Arai, K.; Araya, M. C.; Arceneaux, C. C.; Areeda, J. S.; Arnaud, N.; Arun, K. G.; Ascenzi, S.; Ashton, G.; Ast, M.; Aston, S. M.; Astone, P.; Aufmuth, P.; Aulbert, C.; Babak, S.; Bacon, P.; Bader, M. K. M.; Baker, P. T.; Baldaccini, F.; Ballardin, G.; Ballmer, S. W.; Barayoga, J. C.; Barclay, S. E.; Barish, B. C.; Barker, D.; Barone, F.; Barr, B.; Barsotti, L.; Barsuglia, M.; Barta, D.; Bartlett, J.; Bartos, I.; Bassiri, R.; Basti, A.; Batch, J. C.; Baune, C.; Bavigadda, V.; Bazzan, M.; Bejger, M.; Bell, A. S.; Berger, B. K.; Bergmann, G.; Berry, C. P. L.; Bersanetti, D.; Bertolini, A.; Betzwieser, J.; Bhagwat, S.; Bhandare, R.; Bilenko, I. A.; Billingsley, G.; Birch, J.; Birney, R.; Biscans, S.; Bisht, A.; Bitossi, M.; Biwer, C.; Bizouard, M. A.; Blackburn, J. K.; Blair, C. D.; Blair, D. G.; Blair, R. M.; Bloemen, S.; Bock, O.; Boer, M.; Bogaert, G.; Bogan, C.; Bohe, A.; Bond, C.; Bondu, F.; Bonnand, R.; Boom, B. A.; Bork, R.; Boschi, V.; Bose, S.; Bouffanais, Y.; Bozzi, A.; Bradaschia, C.; Brady, P. R.; Braginsky, V. B.; Branchesi, M.; Brau, J. E.; Briant, T.; Brillet, A.; Brinkmann, M.; Brisson, V.; Brockill, P.; Broida, J. E.; Brooks, A. F.; Brown, D. A.; Brown, D. D.; Brown, N. M.; Brunett, S.; Buchanan, C. C.; Buikema, A.; Bulik, T.; Bulten, H. J.; Buonanno, A.; Buskulic, D.; Buy, C.; Byer, R. L.; Cabero, M.; Cadonati, L.; Cagnoli, G.; Cahillane, C.; Calderón Bustillo, J.; Callister, T.; Calloni, E.; Camp, J. B.; Cannon, K. C.; Cao, J.; Capano, C. D.; Capocasa, E.; Carbognani, F.; Caride, S.; Casanueva Diaz, J.; Casentini, C.; Caudill, S.; Cavaglià, M.; Cavalier, F.; Cavalieri, R.; Cella, G.; Cepeda, C. B.; Cerboni Baiardi, L.; Cerretani, G.; Cesarini, E.; Chamberlin, S. J.; Chan, M.; Chao, S.; Charlton, P.; Chassande-Mottin, E.; Cheeseboro, B. D.; Chen, H. Y.; Chen, Y.; Cheng, C.; Chincarini, A.; Chiummo, A.; Cho, H. S.; Cho, M.; Chow, J. H.; Christensen, N.; Chu, Q.; Chua, S.; Chung, S.; Ciani, G.; Clara, F.; Clark, J. A.; Cleva, F.; Coccia, E.; Cohadon, P.-F.; Colla, A.; Collette, C. G.; Cominsky, L.; Constancio, M.; Conte, A.; Conti, L.; Cook, D.; Corbitt, T. R.; Cornish, N.; Corsi, A.; Cortese, S.; Costa, C. A.; Coughlin, M. W.; Coughlin, S. B.; Coulon, J.-P.; Countryman, S. T.; Couvares, P.; Cowan, E. E.; Coward, D. M.; Cowart, M. J.; Coyne, D. C.; Coyne, R.; Craig, K.; Creighton, J. D. E.; Cripe, J.; Crowder, S. G.; Cumming, A.; Cunningham, L.; Cuoco, E.; Dal Canton, T.; Danilishin, S. L.; D'Antonio, S.; Danzmann, K.; Darman, N. S.; Dasgupta, A.; Da Silva Costa, C. F.; Dattilo, V.; Dave, I.; Davier, M.; Davies, G. S.; Daw, E. J.; Day, R.; De, S.; DeBra, D.; Debreczeni, G.; Degallaix, J.; De Laurentis, M.; Deléglise, S.; Del Pozzo, W.; Denker, T.; Dent, T.; Dergachev, V.; De Rosa, R.; DeRosa, R. T.; DeSalvo, R.; Devine, R. C.; Dhurandhar, S.; Díaz, M. C.; Di Fiore, L.; Di Giovanni, M.; Di Girolamo, T.; Di Lieto, A.; Di Pace, S.; Di Palma, I.; Di Virgilio, A.; Dolique, V.; Donovan, F.; Dooley, K. L.; Doravari, S.; Douglas, R.; Downes, T. P.; Drago, M.; Drever, R. W. P.; Driggers, J. C.; Ducrot, M.; Dwyer, S. E.; Edo, T. B.; Edwards, M. C.; Effler, A.; Eggenstein, H.-B.; Ehrens, P.; Eichholz, J.; Eikenberry, S. S.; Engels, W.; Essick, R. C.; Etzel, T.; Evans, M.; Evans, T. M.; Everett, R.; Factourovich, M.; Fafone, V.; Fair, H.; Fan, X.; Fang, Q.; Farinon, S.; Farr, B.; Farr, W. M.; Favata, M.; Fays, M.; Fehrmann, H.; Fejer, M. M.; Fenyvesi, E.; Ferrante, I.; Ferreira, E. C.; Ferrini, F.; Fidecaro, F.; Fiori, I.; Fiorucci, D.; Fisher, R. P.; Flaminio, R.; Fletcher, M.; Fournier, J.-D.; Frasca, S.; Frasconi, F.; Frei, Z.; Freise, A.; Frey, R.; Frey, V.; Fritschel, P.; Frolov, V. V.; Fulda, P.; Fyffe, M.; Gabbard, H. A. G.; Gair, J. R.; Gammaitoni, L.; Gaonkar, S. G.; Garufi, F.; Gaur, G.; Gehrels, N.; Gemme, G.; Geng, P.; Genin, E.; Gennai, A.; George, J.; Gergely, L.; Germain, V.; Ghosh, Abhirup; Ghosh, Archisman; Ghosh, S.; Giaime, J. A.; Giardina, K. D.; Giazotto, A.; Gill, K.; Glaefke, A.; Goetz, E.; Goetz, R.; Gondan, L.; González, G.; Gonzalez Castro, J. M.; Gopakumar, A.; Gordon, N. A.; Gorodetsky, M. L.; Gossan, S. E.; Gosselin, M.; Gouaty, R.; Grado, A.; Graef, C.; Graff, P. B.; Granata, M.; Grant, A.; Gras, S.; Gray, C.; Greco, G.; Green, A. C.; Groot, P.; Grote, H.; Grunewald, S.; Guidi, G. M.; Guo, X.; Gupta, A.; Gupta, M. K.; Gushwa, K. E.; Gustafson, E. K.; Gustafson, R.; Hacker, J. J.; Hall, B. R.; Hall, E. D.; Hammond, G.; Haney, M.; Hanke, M. M.; Hanks, J.; Hanna, C.; Hanson, J.; Hardwick, T.; Harms, J.; Harry, G. M.; Harry, I. W.; Hart, M. J.; Hartman, M. T.; Haster, C.-J.; Haughian, K.; Heidmann, A.; Heintze, M. C.; Heitmann, H.; Hello, P.; Hemming, G.; Hendry, M.; Heng, I. S.; Hennig, J.; Henry, J.; Heptonstall, A. W.; Heurs, M.; Hild, S.; Hoak, D.; Hofman, D.; Holt, K.; Holz, D. E.; Hopkins, P.; Hough, J.; Houston, E. A.; Howell, E. J.; Hu, Y. M.; Huang, S.; Huerta, E. A.; Huet, D.; Hughey, B.; Huttner, S. H.; Huynh-Dinh, T.; Indik, N.; Ingram, D. R.; Inta, R.; Isa, H. N.; Isac, J.-M.; Isi, M.; Isogai, T.; Iyer, B. R.; Izumi, K.; Jacqmin, T.; Jang, H.; Jani, K.; Jaranowski, P.; Jawahar, S.; Jian, L.; Jiménez-Forteza, F.; Johnson, W. W.; Jones, D. I.; Jones, R.; Jonker, R. J. G.; Ju, L.; Haris, K.; Kalaghatgi, C. V.; Kalogera, V.; Kandhasamy, S.; Kang, G.; Kanner, J. B.; Kapadia, S. J.; Karki, S.; Karvinen, K. S.; Kasprzack, M.; Katsavounidis, E.; Katzman, W.; Kaufer, S.; Kaur, T.; Kawabe, K.; Kéfélian, F.; Kehl, M. S.; Keitel, D.; Kelley, D. B.; Kells, W.; Kennedy, R.; Key, J. S.; Khalili, F. Y.; Khan, I.; Khan, Z.; Khazanov, E. A.; Kijbunchoo, N.; Kim, Chi-Woong; Kim, Chunglee; Kim, J.; Kim, K.; Kim, N.; Kim, W.; Kim, Y.-M.; Kimbrell, S. J.; King, E. J.; King, P. J.; Kissel, J. S.; Klein, B.; Kleybolte, L.; Klimenko, S.; Koehlenbeck, S. M.; Koley, S.; Kondrashov, V.; Kontos, A.; Korobko, M.; Korth, W. Z.; Kowalska, I.; Kozak, D. B.; Kringel, V.; Królak, A.; Krueger, C.; Kuehn, G.; Kumar, P.; Kumar, R.; Kuo, L.; Kutynia, A.; Lackey, B. D.; Landry, M.; Lange, J.; Lantz, B.; Lasky, P. D.; Laxen, M.; Lazzarini, A.; Lazzaro, C.; Leaci, P.; Leavey, S.; Lebigot, E. O.; Lee, C. H.; Lee, H. K.; Lee, H. M.; Lee, K.; Lenon, A.; Leonardi, M.; Leong, J. R.; Leroy, N.; Letendre, N.; Levin, Y.; Lewis, J. B.; Li, T. G. F.; Libson, A.; Littenberg, T. B.; Lockerbie, N. A.; Lombardi, A. L.; Lord, J. E.; Lorenzini, M.; Loriette, V.; Lormand, M.; Losurdo, G.; Lough, J. D.; Lück, H.; Lundgren, A. P.; Lynch, R.; Ma, Y.; Machenschalk, B.; MacInnis, M.; Macleod, D. M.; Magaña-Sandoval, F.; Zertuche, L. Magaña; Magee, R. M.; Majorana, E.; Maksimovic, I.; Malvezzi, V.; Man, N.; Mandic, V.; Mangano, V.; Mansell, G. L.; Manske, M.; Mantovani, M.; Marchesoni, F.; Marion, F.; Márka, S.; Márka, Z.; Markosyan, A. S.; Maros, E.; Martelli, F.; Martellini, L.; Martin, I. W.; Martynov, D. V.; Marx, J. N.; Mason, K.; Masserot, A.; Massinger, T. J.; Masso-Reid, M.; Mastrogiovanni, S.; Matichard, F.; Matone, L.; Mavalvala, N.; Mazumder, N.; McCarthy, R.; McClelland, D. E.; McCormick, S.; McGuire, S. C.; McIntyre, G.; McIver, J.; McManus, D. J.; McRae, T.; McWilliams, S. T.; Meacher, D.; Meadors, G. D.; Meidam, J.; Melatos, A.; Mendell, G.; Mercer, R. A.; Merilh, E. L.; Merzougui, M.; Meshkov, S.; Messenger, C.; Messick, C.; Metzdorff, R.; Meyers, P. M.; Mezzani, F.; Miao, H.; Michel, C.; Middleton, H.; Mikhailov, E. E.; Milano, L.; Miller, A. L.; Miller, A.; Miller, B. B.; Miller, J.; Millhouse, M.; Minenkov, Y.; Ming, J.; Mirshekari, S.; Mishra, C.; Mitra, S.; Mitrofanov, V. P.; Mitselmakher, G.; Mittleman, R.; Moggi, A.; Mohan, M.; Mohapatra, S. R. P.; Montani, M.; Moore, B. C.; Moore, C. J.; Moraru, D.; Moreno, G.; Morriss, S. R.; Mossavi, K.; Mours, B.; Mow-Lowry, C. M.; Mueller, G.; Muir, A. W.; Mukherjee, Arunava; Mukherjee, D.; Mukherjee, S.; Mukund, N.; Mullavey, A.; Munch, J.; Murphy, D. J.; Murray, P. G.; Mytidis, A.; Nardecchia, I.; Naticchioni, L.; Nayak, R. K.; Nedkova, K.; Nelemans, G.; Nelson, T. J. N.; Neri, M.; Neunzert, A.; Newton, G.; Nguyen, T. T.; Nielsen, A. B.; Nissanke, S.; Nitz, A.; Nocera, F.; Nolting, D.; Normandin, M. E. N.; Nuttall, L. K.; Oberling, J.; Ochsner, E.; O'Dell, J.; Oelker, E.; Ogin, G. H.; Oh, J. J.; Oh, S. H.; Ohme, F.; Oliver, M.; Oppermann, P.; Oram, Richard J.; O'Reilly, B.; O'Shaughnessy, R.; Ottaway, D. J.; Overmier, H.; Owen, B. J.; Pai, A.; Pai, S. A.; Palamos, J. R.; Palashov, O.; Palomba, C.; Pal-Singh, A.; Pan, H.; Pankow, C.; Pant, B. C.; Paoletti, F.; Paoli, A.; Papa, M. A.; Paris, H. R.; Parker, W.; Pascucci, D.; Pasqualetti, A.; Passaquieti, R.; Passuello, D.; Patricelli, B.; Patrick, Z.; Pearlstone, B. L.; Pedraza, M.; Pedurand, R.; Pekowsky, L.; Pele, A.; Penn, S.; Perreca, A.; Perri, L. M.; Phelps, M.; Piccinni, O. J.; Pichot, M.; Piergiovanni, F.; Pierro, V.; Pillant, G.; Pinard, L.; Pinto, I. M.; Pitkin, M.; Poe, M.; Poggiani, R.; Popolizio, P.; Post, A.; Powell, J.; Prasad, J.; Predoi, V.; Prestegard, T.; Price, L. R.; Prijatelj, M.; Principe, M.; Privitera, S.; Prodi, G. A.; Prokhorov, L.; Puncken, O.; Punturo, M.; Puppo, P.; Pürrer, M.; Qi, H.; Qin, J.; Qiu, S.; Quetschke, V.; Quintero, E. A.; Quitzow-James, R.; Raab, F. J.; Rabeling, D. S.; Radkins, H.; Raffai, P.; Raja, S.; Rajan, C.; Rakhmanov, M.; Rapagnani, P.; Raymond, V.; Razzano, M.; Re, V.; Read, J.; Reed, C. M.; Regimbau, T.; Rei, L.; Reid, S.; Reitze, D. H.; Rew, H.; Reyes, S. D.; Ricci, F.; Riles, K.; Rizzo, M.; Robertson, N. A.; Robie, R.; Robinet, F.; Rocchi, A.; Rolland, L.; Rollins, J. G.; Roma, V. J.; Romano, J. D.; Romano, R.; Romanov, G.; Romie, J. H.; Rosińska, D.; Rowan, S.; Rüdiger, A.; Ruggi, P.; Ryan, K.; Sachdev, S.; Sadecki, T.; Sadeghian, L.; Sakellariadou, M.; Salconi, L.; Saleem, M.; Salemi, F.; Samajdar, A.; Sammut, L.; Sanchez, E. J.; Sandberg, V.; Sandeen, B.; Sanders, J. R.; Sassolas, B.; Saulson, P. R.; Sauter, O. E. S.; Savage, R. L.; Sawadsky, A.; Schale, P.; Schilling, R.; Schmidt, J.; Schmidt, P.; Schnabel, R.; Schofield, R. M. S.; Schönbeck, A.; Schreiber, E.; Schuette, D.; Schutz, B. F.; Scott, J.; Scott, S. M.; Sellers, D.; Sengupta, A. S.; Sentenac, D.; Sequino, V.; Sergeev, A.; Setyawati, Y.; Shaddock, D. A.; Shaffer, T.; Shahriar, M. S.; Shaltev, M.; Shapiro, B.; Shawhan, P.; Sheperd, A.; Shoemaker, D. H.; Siellez, K.; Siemens, X.; Sieniawska, M.; Sigg, D.; Silva, A. D.; Singer, A.; Singer, L. P.; Singh, A.; Singh, R.; Singhal, A.; Sintes, A. M.; Slagmolen, B. J. J.; Smith, J. R.; Smith, N. D.; Smith, R. J. E.; Son, E. J.; Sorazu, B.; Sorrentino, F.; Souradeep, T.; Srivastava, A. K.; Staley, A.; Steinke, M.; Steinlechner, J.; Steinlechner, S.; Steinmeyer, D.; Stephens, B. C.; Stone, R.; Strain, K. A.; Straniero, N.; Stratta, G.; Strauss, N. A.; Strigin, S.; Sturani, R.; Stuver, A. L.; Summerscales, T. Z.; Sun, L.; Sunil, S.; Sutton, P. J.; Swinkels, B. L.; Szczepańczyk, M. J.; Tacca, M.; Talukder, D.; Tanner, D. B.; Tápai, M.; Tarabrin, S. P.; Taracchini, A.; Taylor, R.; Theeg, T.; Thirugnanasambandam, M. P.; Thomas, E. G.; Thomas, M.; Thomas, P.; Thorne, K. A.; Thorne, K. S.; Thrane, E.; Tiwari, S.; Tiwari, V.; Tokmakov, K. V.; Toland, K.; Tomlinson, C.; Tonelli, M.; Tornasi, Z.; Torres, C. V.; Torrie, C. I.; Töyrä, D.; Travasso, F.; Traylor, G.; Trifirò, D.; Tringali, M. C.; Trozzo, L.; Tse, M.; Turconi, M.; Tuyenbayev, D.; Ugolini, D.; Unnikrishnan, C. S.; Urban, A. L.; Usman, S. A.; Vahlbruch, H.; Vajente, G.; Valdes, G.; van Bakel, N.; van Beuzekom, M.; van den Brand, J. F. J.; Van Den Broeck, C.; Vander-Hyde, D. C.; van der Schaaf, L.; van Heijningen, J. V.; van Veggel, A. A.; Vardaro, M.; Vass, S.; Vasúth, M.; Vaulin, R.; Vecchio, A.; Vedovato, G.; Veitch, J.; Veitch, P. J.; Venkateswara, K.; Verkindt, D.; Vetrano, F.; Viceré, A.; Vinciguerra, S.; Vine, D. J.; Vinet, J.-Y.; Vitale, S.; Vo, T.; Vocca, H.; Vorvick, C.; Voss, D. V.; Vousden, W. D.; Vyatchanin, S. P.; Wade, A. R.; Wade, L. E.; Wade, M.; Walker, M.; Wallace, L.; Walsh, S.; Wang, G.; Wang, H.; Wang, M.; Wang, X.; Wang, Y.; Ward, R. L.; Warner, J.; Was, M.; Weaver, B.; Wei, L.-W.; Weinert, M.; Weinstein, A. J.; Weiss, R.; Wen, L.; Weßels, P.; Westphal, T.; Wette, K.; Whelan, J. T.; Whiting, B. F.; Williams, R. D.; Williamson, A. R.; Willis, J. L.; Willke, B.; Wimmer, M. H.; Winkler, W.; Wipf, C. C.; Wittel, H.; Woan, G.; Woehler, J.; Worden, J.; Wright, J. L.; Wu, D. S.; Wu, G.; Yablon, J.; Yam, W.; Yamamoto, H.; Yancey, C. C.; Yu, H.; Yvert, M.; ZadroŻny, A.; Zangrando, L.; Zanolin, M.; Zendri, J.-P.; Zevin, M.; Zhang, L.; Zhang, M.; Zhang, Y.; Zhao, C.; Zhou, M.; Zhou, Z.; Zhu, X. J.; Zucker, M. E.; Zuraw, S. E.; Zweizig, J.; Boyle, M.; Campanelli, M.; Chu, T.; Clark, M.; Fauchon-Jones, E.; Fong, H.; Healy, J.; Hemberger, D.; Hinder, I.; Husa, S.; Kalaghati, C.; Khan, S.; Kidder, L. E.; Kinsey, M.; Laguna, P.; London, L. T.; Lousto, C. O.; Lovelace, G.; Ossokine, S.; Pannarale, F.; Pfeiffer, H. P.; Scheel, M.; Shoemaker, D. M.; Szilagyi, B.; Teukolsky, S.; Vinuales, A. Vano; Zlochower, Y.; LIGO Scientific Collaboration; Virgo Collaboration
2016-09-01
We compare GW150914 directly to simulations of coalescing binary black holes in full general relativity, including several performed specifically to reproduce this event. Our calculations go beyond existing semianalytic models, because for all simulations—including sources with two independent, precessing spins—we perform comparisons which account for all the spin-weighted quadrupolar modes, and separately which account for all the quadrupolar and octopolar modes. Consistent with the posterior distributions reported by Abbott et al. [Phys. Rev. Lett. 116, 241102 (2016)] (at the 90% credible level), we find the data are compatible with a wide range of nonprecessing and precessing simulations. Follow-up simulations performed using previously estimated binary parameters most resemble the data, even when all quadrupolar and octopolar modes are included. Comparisons including only the quadrupolar modes constrain the total redshifted mass Mz∈[64 M⊙-82 M⊙] , mass ratio 1 /q =m2/m1∈[0.6 ,1 ], and effective aligned spin χeff∈[-0.3 ,0.2 ], where χeff=(S1/m1+S2/m2).L ^/M . Including both quadrupolar and octopolar modes, we find the mass ratio is even more tightly constrained. Even accounting for precession, simulations with extreme mass ratios and effective spins are highly inconsistent with the data, at any mass. Several nonprecessing and precessing simulations with similar mass ratio and χeff are consistent with the data. Though correlated, the components' spins (both in magnitude and directions) are not significantly constrained by the data: the data is consistent with simulations with component spin magnitudes a1 ,2 up to at least 0.8, with random orientations. Further detailed follow-up calculations are needed to determine if the data contain a weak imprint from transverse (precessing) spins. For nonprecessing binaries, interpolating between simulations, we reconstruct a posterior distribution consistent with previous results. The final black hole
FASTREACT – An efficient numerical framework for the solution of reactive transport problems
International Nuclear Information System (INIS)
Trinchero, Paolo; Molinero, Jorge; Román-Ross, Gabriela; Berglund, Sten; Selroos, Jan-Olof
2014-01-01
Highlights: • We present a tool for the efficient solution of reactive transport problems. • The tool is used to simulate radionuclide transport in a two-dimensional medium. • The results are successfully compared with those obtained using an Eulerian approach. • A large-scale application example is also solved. • The results show that the proposed tool can efficiently solve large-scale models. - Abstract: In the framework of safety assessment studies for geological disposal, large scale reactive transport models are powerful inter-disciplinary tools aiming at supporting regulatory decision making as well as providing input to repository engineering activities. Important aspects of these kinds of models are their often very large temporal and spatial modelling scales and the need to integrate different non-linear processes (e.g., mineral dissolution and precipitation, adsorption and desorption, microbial reactions and redox transformations). It turns out that these types of models may be computationally highly demanding. In this work, we present a Lagrangian-based framework, denoted as FASTREACT, that aims at solving multi-component-reactive transport problems with a computationally efficient approach allowing complex modelling problems to be solved in large spatial and temporal scales. The tool has been applied to simulate radionuclide migration in a synthetic heterogeneous transmissivity field and the results have been successfully compared with those obtained using a standard Eulerian approach. Finally, the same geochemical model has been coupled to an ensemble of realistic three-dimensional transport pathways to simulate the migration of a set of radionuclides from a hypothetical repository for spent nuclear fuel to the surface. The results of this modelling exercise, which includes key processes such as the exchange of mass between the conductive fractures and the matrix, show that FASTREACT can efficiently solve large-scale reactive transport models
Directory of Open Access Journals (Sweden)
Maciejewska Beata
2017-01-01
Full Text Available This paper shows the results concerning flow boiling heat transfer in an asymmetrically heated vertical minichannel. The heated element for FC-72 Fluorinert flowing in that minichannel was a thin foil. The foil surface temperature was monitored continuously at 18 points by K-type thermocouples from the outer foil surface. Fluid temperature and pressure in the minichannel inlet and outlet, current supplied to the foil and voltage drop were also monitored. Measurements were carried out at 1 s intervals. The objective was to determine the heat transfer coefficient on the heated foil–fluid contact surface in the minichannel. It was obtained from the Robin boundary condition. The foil temperature was the result of solving the nonstationary two-dimensional inverse boundary problem in the heated foil. Using the FEM combined with Trefftz functions as basis functions solved the problem. The unknown temperature values at nodes were calculated by minimising the adequate functional. The values of local heat transfer coefficients were consistent with the results obtained by the authors in their previous studies when steady-state conditions were analysed. This time, however, these values were analysed as time dependent, which facilitated observation of coefficient changes that were impossible to observe under the steady-state conditions.
Nonlinear Dispersive Elastic Waves in Solids: Exact, Approximate, and Numerical Solutions
Khajehtourian, Romik
Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat, or fluid flow are all likely to involve wave dynamics at some level. A particular class of problems is concerned with the propagation of elastic waves in a solid medium, such as a fiber-reinforced composite material responding to vibratory excitations, or soil and rock admitting seismic waves moments after the onset of an earthquake, or phonon transport in a semiconducting crystal like silicon. Regardless of the type of wave, the dispersion relation provides a fundamental characterization of the elastodynamic properties of the medium. The first part of the dissertation examines the propagation of a large-amplitude elastic wave in a one-dimensional homogeneous medium with a focus on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, an exact, closed-form formulation is presented for the treatment of two types of nonlinearity in the strain-displacement gradient relation: Green-Lagrange and Hencky. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformation). A high-order perturbation analysis is also conducted yielding an explicit analytical space-time solution, which is shown to be spectrally accurate. The results establish a perfect match between theory and simulation and reveal that regardless of the strength of the nonlinearity, the dispersion relation fully embodies all information pertaining to the nonlinear harmonic generation mechanism that unfolds as an arbitrary-profiled wave evolves in the medium. In the second part of the dissertation, the analysis is extended to a continuous periodic thin rod exhibiting multiple phases or embedded local resonators. The
Sun, Shuyu; Salama, Amgad; El-Amin, Mohamed
2012-01-01
A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.
International Nuclear Information System (INIS)
Khazanov, G.V.; Koen, M.A.; Burenkov, S.I.
1979-01-01
Considered is the dinamics of photoelectron fluxes formation in the Earth plasmasphere with account of zone interaction of free and trapped photoelectrons. An algorithm and the results of numerical solution of the equation are presented. The problem of boundary condition choice is discussed. The angular distribution of 10 eV energy photoelectrons at different altitudes of plasmasphere is presented as an example. It is shown that the changes of photoelectron distribution function from bottom of plasmasphere to the top of a force line of the geomagnetic field are within the 1.6 limits. Presented is the estimate of plasmasphere transmittance value and its comparison with the experiment for Mc Ilwain parameter L=2
Sun, Shuyu
2012-06-02
A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.
International Nuclear Information System (INIS)
Li, X.; Raghuveer, M.R.; Ciric, I.R.
1997-01-01
Three finite element numerical algorithms were developed for solving the ionized field of unipolar HVDC transmission lines in the presence of wind. The solution of the ionized field was derived by transforming the boundary value problem to an optimization problem. The Gaussian iterative method was used for all three algorithms. The main features of each algorithm were presented and compared. The advantage of the first algorithm is that there is no need to assume the space charge density to be uniform on the conductor surface and therefore it is flexible in dealing with a complicated geometry like that of a bundled conductor. The main features of the other two algorithms were their ability to perform better with increase in wind velocity. Algorithm No. 2 was better than algorithm No. 1 in terms of storage and computational time. Algorithm No. 3 also had the advantage of using a triangular fine volume method which allowed computations to proceed faster. 11 refs., 10 figs
International Nuclear Information System (INIS)
Yudov, Y.V.
2001-01-01
The functional part of the KORSAR computer code is based on the computational unit for the reactor system thermal-hydraulics and other thermal power systems with water cooling. The two-phase flow dynamics of the thermal-hydraulic network is modelled by KORSAR in one-dimensional two-fluid (non-equilibrium and nonhomogeneous) approximation with the same pressure of both phases. Each phase is characterized by parameters averaged over the channel sections, and described by the conservation equations for mass, energy and momentum. The KORSAR computer code relies upon a novel approach to mathematical modelling of two-phase dispersed-annular flows. This approach allows a two-fluid model to differentiate the effects of the liquid film and droplets in the gas core on the flow characteristics. A semi-implicit numerical scheme has been chosen for deriving discrete analogs the conservation equations in KORSAR. In the semi-implicit numerical scheme, solution of finite-difference equations is reduced to the problem of determining the pressure field at a new time level. For the one-channel case, the pressure field is found from the solution of a system of linear algebraic equations by using the tri-diagonal matrix method. In the branched network calculation, the matrix of coefficients in the equations describing the pressure field is no longer tri-diagonal but has a sparseness structure. In this case, the system of linear equations for the pressure field can be solved with any of the known classical methods. Such an approach is implemented in the existing best-estimate thermal-hydraulic computer codes (TRAC, RELAP5, etc.) For the KORSAR computer code, we have developed a new non-iterative method for calculating the pressure field in the network of any topology. This method is based on the tri-diagonal matrix method and performs well when solving the thermal-hydraulic network problems. (author)
Dehghan, Mehdi; Mohammadi, Vahid
2017-03-01
As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank-Nicolson scheme and another one is based on explicit Runge-Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.
Bower, Dan J.; Sanan, Patrick; Wolf, Aaron S.
2018-01-01
The energy balance of a partially molten rocky planet can be expressed as a non-linear diffusion equation using mixing length theory to quantify heat transport by both convection and mixing of the melt and solid phases. Crucially, in this formulation the effective or eddy diffusivity depends on the entropy gradient, ∂S / ∂r , as well as entropy itself. First we present a simplified model with semi-analytical solutions that highlights the large dynamic range of ∂S / ∂r -around 12 orders of magnitude-for physically-relevant parameters. It also elucidates the thermal structure of a magma ocean during the earliest stage of crystal formation. This motivates the development of a simple yet stable numerical scheme able to capture the large dynamic range of ∂S / ∂r and hence provide a flexible and robust method for time-integrating the energy equation. Using insight gained from the simplified model, we consider a full model, which includes energy fluxes associated with convection, mixing, gravitational separation, and conduction that all depend on the thermophysical properties of the melt and solid phases. This model is discretised and evolved by applying the finite volume method (FVM), allowing for extended precision calculations and using ∂S / ∂r as the solution variable. The FVM is well-suited to this problem since it is naturally energy conserving, flexible, and intuitive to incorporate arbitrary non-linear fluxes that rely on lookup data. Special attention is given to the numerically challenging scenario in which crystals first form in the centre of a magma ocean. The computational framework we devise is immediately applicable to modelling high melt fraction phenomena in Earth and planetary science research. Furthermore, it provides a template for solving similar non-linear diffusion equations that arise in other science and engineering disciplines, particularly for non-linear functional forms of the diffusion coefficient.
International Nuclear Information System (INIS)
Fronteau, J.; Combis, P.
1984-08-01
A Lagrangian method is introduced for the integration of non-linear Fokker-Planck equations. Examples of exact solutions obtained in this way are given, and also the explicit scheme used for the computation of numerical solutions. The method is, in addition, shown to be of a Lie-admissible type
Sun, Shuyu
2012-09-01
In this paper we introduce a new technique for the numerical solution of the various partial differential equations governing flow and transport phenomena in porous media. This method is proposed to be used in high level programming languages like MATLAB, Python, etc., which show to be more efficient for certain mathematical operations than for others. The proposed technique utilizes those operations in which these programming languages are efficient the most and keeps away as much as possible from those inefficient, time-consuming operations. In particular, this technique is based on the minimization of using multiple indices looping operations by reshaping the unknown variables into one-dimensional column vectors and performing the numerical operations using shifting matrices. The cell-centered information as well as the face-centered information are shifted to the adjacent face-center and cell-center, respectively. This enables the difference equations to be done for all the cells at once using matrix operations rather than within loops. Furthermore, for results post-processing, the face-center information can further be mapped to the physical grid nodes for contour plotting and stream lines constructions. In this work we apply this technique to flow and transport phenomena in porous media. © 2012 Elsevier Ltd.
Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.
2012-10-01
A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
Directory of Open Access Journals (Sweden)
Donghan Lee
2013-09-01
Full Text Available Nuclear Magnetic Resonance (NMR spectroscopy is a powerful tool that has enabled experimentalists to characterize molecular dynamics and kinetics spanning a wide range of time-scales from picoseconds to days. This review focuses on addressing the previously inaccessible supra-τc window (defined as τc < supra-τc < 40 μs; in which τc is the overall tumbling time of a molecule from the perspective of local inter-nuclear vector dynamics extracted from residual dipolar couplings (RDCs and from the perspective of conformational exchange captured by relaxation dispersion measurements (RD. The goal of the first section is to present a detailed analysis of how to extract protein dynamics encoded in RDCs and how to relate this information to protein functionality within the previously inaccessible supra-τc window. In the second section, the current state of the art for RD is analyzed, as well as the considerable progress toward pushing the sensitivity of RD further into the supra-τc scale by up to a factor of two (motion up to 25 ms. From the data obtained with these techniques and methodology, the importance of the supra-τ c scale for protein function and molecular recognition is becoming increasingly clearer as the connection between motion on the supra-τc scale and protein functionality from the experimental side is further strengthened with results from molecular dynamics simulations.
Groves, Curtis E.; Ilie, marcel; Shallhorn, Paul A.
2014-01-01
Computational Fluid Dynamics (CFD) is the standard numerical tool used by Fluid Dynamists to estimate solutions to many problems in academia, government, and industry. CFD is known to have errors and uncertainties and there is no universally adopted method to estimate such quantities. This paper describes an approach to estimate CFD uncertainties strictly numerically using inputs and the Student-T distribution. The approach is compared to an exact analytical solution of fully developed, laminar flow between infinite, stationary plates. It is shown that treating all CFD input parameters as oscillatory uncertainty terms coupled with the Student-T distribution can encompass the exact solution.
Hajipour, Mojtaba; Jajarmi, Amin
2018-02-01
Using the Pontryagin's maximum principle for a time-delayed optimal control problem results in a system of coupled two-point boundary-value problems (BVPs) involving both time-advance and time-delay arguments. The analytical solution of this advance-delay two-point BVP is extremely difficult, if not impossible. This paper provides a discrete general form of the numerical solution for the derived advance-delay system by applying a finite difference θ-method. This method is also implemented for the infinite-time horizon time-delayed optimal control problems by using a piecewise version of the θ-method. A matrix formulation and the error analysis of the suggested technique are provided. The new scheme is accurate, fast and very effective for the optimal control of linear and nonlinear time-delay systems. Various types of finite- and infinite-time horizon problems are included to demonstrate the accuracy, validity and applicability of the new technique.
Ha, Sanghyun; Park, Junshin; You, Donghyun
2017-11-01
Utility of the computational power of modern Graphics Processing Units (GPUs) is elaborated for solutions of incompressible Navier-Stokes equations which are integrated using a semi-implicit fractional-step method. Due to its serial and bandwidth-bound nature, the present choice of numerical methods is considered to be a good candidate for evaluating the potential of GPUs for solving Navier-Stokes equations using non-explicit time integration. An efficient algorithm is presented for GPU acceleration of the Alternating Direction Implicit (ADI) and the Fourier-transform-based direct solution method used in the semi-implicit fractional-step method. OpenMP is employed for concurrent collection of turbulence statistics on a CPU while Navier-Stokes equations are computed on a GPU. Extension to multiple NVIDIA GPUs is implemented using NVLink supported by the Pascal architecture. Performance of the present method is experimented on multiple Tesla P100 GPUs compared with a single-core Xeon E5-2650 v4 CPU in simulations of boundary-layer flow over a flat plate. Supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (Ministry of Science, ICT and Future Planning NRF-2016R1E1A2A01939553, NRF-2014R1A2A1A11049599, and Ministry of Trade, Industry and Energy 201611101000230).
International Nuclear Information System (INIS)
Piran, T.
1982-01-01
There are many recent developments in numerical relativity, but there remain important unsolved theoretical and practical problems. The author reviews existing numerical approaches to solution of the exact Einstein equations. A framework for classification and comparison of different numerical schemes is presented. Recent numerical codes are compared using this framework. The discussion focuses on new developments and on currently open questions, excluding a review of numerical techniques. (Auth.)
Ban, David; Sabo, T Michael; Griesinger, Christian; Lee, Donghan
2013-09-26
Nuclear Magnetic Resonance (NMR) spectroscopy is a powerful tool that has enabled experimentalists to characterize molecular dynamics and kinetics spanning a wide range of time-scales from picoseconds to days. This review focuses on addressing the previously inaccessible supra-tc window (defined as τ(c) supra-τ(c) supra-τ(c) window. In the second section, the current state of the art for RD is analyzed, as well as the considerable progress toward pushing the sensitivity of RD further into the supra-τ(c) scale by up to a factor of two (motion up to 25 μs). From the data obtained with these techniques and methodology, the importance of the supra-τ(c) scale for protein function and molecular recognition is becoming increasingly clearer as the connection between motion on the supra-τ(c) scale and protein functionality from the experimental side is further strengthened with results from molecular dynamics simulations.
Yu, Ming-Xiao; Tian, Bo; Chai, Jun; Yin, Hui-Min; Du, Zhong
2017-10-01
In this paper, we investigate a nonlinear fiber described by a (2+1)-dimensional complex Ginzburg-Landau equation with the chromatic dispersion, optical filtering, nonlinear and linear gain. Bäcklund transformation in the bilinear form is constructed. With the modified bilinear method, analytic soliton solutions are obtained. For the soliton, the amplitude can decrease or increase when the absolute value of the nonlinear or linear gain is enlarged, and the width can be compressed or amplified when the absolute value of the chromatic dispersion or optical filtering is enhanced. We study the stability of the numerical solutions numerically by applying the increasing amplitude, embedding the white noise and adding the Gaussian pulse to the initial values based on the analytic solutions, which shows that the numerical solutions are stable, not influenced by the finite initial perturbations.
Russo, David
2017-11-01
The main goal of this study was to test the capability of irrigation water-based and soil-based approaches to control nitrate and chloride mass fluxes and concentrations below the root zone of agricultural fields irrigated with treated waste water (TWW). Using numerical simulations of flow and transport in relatively a fine-textured, unsaturated, spatially heterogeneous, flow domain, scenarios examined include: (i) irrigating with TWW only (REF); (ii) irrigation water is substituted between TWW and desalinized water (ADW); (iii) soil includes a capillary barrier (CB) and irrigating with TWW only (CB + TWW); and (iv) combination of (ii) and a CB (CB + ADW). Considering groundwater quality protection, plausible goals are: (i) to minimize solute discharges leaving the root zone, and, (ii) to maximize the probability that solute concentrations leaving the root zone will not exceed a prescribed, critical value. Results of the analyses suggest that in the case of a seasonal crop (a corn field) subject to irrigations only, with respect to the first goal, the CB + TWW and CB + ADW scenarios provide similar, excellent results, better than the ADW scenario; with respect to the second goal, however, the CB + ADW scenario gave substantially better results than the CB + TWW scenario. In the case a multiyear, perennial crop (a citrus orchard), subject to a sequence of irrigation and rainfall periods, for both solutes, and, particularly, nitrate, with respect to the two goals, both the ADW and CB + ADW scenarios perform better than the CB + TWW scenario. As compared with the REF and CB + TWW scenarios, the ADW and CB + ADW scenarios substantially reduce nitrogen mass fluxes to the groundwater and to the atmosphere, and, essentially, did not reduce nitrogen mass fluxes to the trees. Similar results, even better, were demonstrated for a relatively coarse-textured, spatially heterogeneous soil.
Energy Technology Data Exchange (ETDEWEB)
Spoerl, Andreas
2008-06-05
Quantum computers are one of the next technological steps in modern computer science. Some of the relevant questions that arise when it comes to the implementation of quantum operations (as building blocks in a quantum algorithm) or the simulation of quantum systems are studied. Numerical results are gathered for variety of systems, e.g. NMR systems, Josephson junctions and others. To study quantum operations (e.g. the quantum fourier transform, swap operations or multiply-controlled NOT operations) on systems containing many qubits, a parallel C++ code was developed and optimised. In addition to performing high quality operations, a closer look was given to the minimal times required to implement certain quantum operations. These times represent an interesting quantity for the experimenter as well as for the mathematician. The former tries to fight dissipative effects with fast implementations, while the latter draws conclusions in the form of analytical solutions. Dissipative effects can even be included in the optimisation. The resulting solutions are relaxation and time optimised. For systems containing 3 linearly coupled spin (1)/(2) qubits, analytical solutions are known for several problems, e.g. indirect Ising couplings and trilinear operations. A further study was made to investigate whether there exists a sufficient set of criteria to identify systems with dynamics which are invertible under local operations. Finally, a full quantum algorithm to distinguish between two knots was implemented on a spin(1)/(2) system. All operations for this experiment were calculated analytically. The experimental results coincide with the theoretical expectations. (orig.)
Ortleb, Sigrun; Seidel, Christian
2017-07-01
In this second symposium at the limits of experimental and numerical methods, recent research is presented on practically relevant problems. Presentations discuss experimental investigation as well as numerical methods with a strong focus on application. In addition, problems are identified which require a hybrid experimental-numerical approach. Topics include fast explicit diffusion applied to a geothermal energy storage tank, noise in experimental measurements of electrical quantities, thermal fluid structure interaction, tensegrity structures, experimental and numerical methods for Chladni figures, optimized construction of hydroelectric power stations, experimental and numerical limits in the investigation of rain-wind induced vibrations as well as the application of exponential integrators in a domain-based IMEX setting.
International Nuclear Information System (INIS)
Cheng, C.Z.
1988-12-01
A nonvariational ideal MHD stability code (NOVA) has been developed. In a general flux coordinate (/psi/, θ, /zeta/) system with an arbitrary Jacobian, the NOVA code employs Fourier expansions in the generalized poloidal angle θ and generalized toroidal angle /zeta/ directions, and cubic-B spline finite elements in the radial /psi/ direction. Extensive comparisons with these variational ideal MHD codes show that the NOVA code converges faster and gives more accurate results. An extended version of NOVA is developed to integrate non-Hermitian eigenmode equations due to energetic particles. The set of non-Hermitian integro-differential eigenmode equations is numerically solved by the NOVA-K code. We have studied the problems of the stabilization of ideal MHD internal kink modes by hot particle pressure and the excitation of ''fishbone'' internal kink modes by resonating with the energetic particle magnetic drift frequency. Comparisons with analytical solutions show that the values of the critical β/sub h/ from the analytical theory can be an order of magnitude different from those computed by the NOVA-K code. 24 refs., 11 figs., 1 tab
Energy Technology Data Exchange (ETDEWEB)
Cheng, C.Z.
1988-12-01
A nonvariational ideal MHD stability code (NOVA) has been developed. In a general flux coordinate (/psi/, theta, /zeta/) system with an arbitrary Jacobian, the NOVA code employs Fourier expansions in the generalized poloidal angle theta and generalized toroidal angle /zeta/ directions, and cubic-B spline finite elements in the radial /psi/ direction. Extensive comparisons with these variational ideal MHD codes show that the NOVA code converges faster and gives more accurate results. An extended version of NOVA is developed to integrate non-Hermitian eigenmode equations due to energetic particles. The set of non-Hermitian integro-differential eigenmode equations is numerically solved by the NOVA-K code. We have studied the problems of the stabilization of ideal MHD internal kink modes by hot particle pressure and the excitation of ''fishbone'' internal kink modes by resonating with the energetic particle magnetic drift frequency. Comparisons with analytical solutions show that the values of the critical ..beta../sub h/ from the analytical theory can be an order of magnitude different from those computed by the NOVA-K code. 24 refs., 11 figs., 1 tab.
Papadopoulos , D. F.; Anastassi , Z. A.; Simos , T. E.
2010-01-01
Abstract A new Runge-Kutta-Nystrom method, with phase-lag and amplification error of order infinity, for the numerical solution of the Schrodinger equation is developed in this paper. The new method is based on the Runge-Kutta-Nystrom method with fourth algebraic order, developed by Dormand, El-Mikkawy and Prince. Numerical illustrations indicate that the new method is much more efficient than other methods derived for the same purpose. phone: +30-210-9421510 (Simos, T. E.) ...
Energy Technology Data Exchange (ETDEWEB)
Prinja, A.K.
1998-09-01
In this work, it has been shown that, for the given sets of parameters (transport coefficients), the Tangent-Predictor (TP) continuation method, which was used in the coarsest grid, works remarkably well. The problems in finding an initial guess that resides well within Newton`s method radius of convergence are alleviated by correcting the initial guess by the predictor step of the TP method. The TP method works well also in neutral gas puffing and impurity simulations. The neutral gas puffing simulation is performed by systematically increasing the fraction of puffing rate according to the TP method until it reaches a desired condition. Similarly, the impurity simulation characterized by using the fraction of impurity density as the continuation parameter, is carried out in line with the TP method. Both methods show, as expected, a better performance than the classical embedding (CE) method. The convergence criteria {epsilon} is set to be 10{sup {minus}9} based on the fact that lower value of {epsilon} does not alter the solution significantly. Correspondingly, the number of Newton`s iterations in the corrector step of the TP method decrease substantially, an extra point in terms of code speed. The success of the TP method enlarges the possibility of including other sets of parameters (operations and physics). With the availability of the converged coarsest grid solution, the next forward step to the multigrid cycle becomes possible. The multigrid method shows that the memory storage problems that plagued the application of Newton`s method on fine grids, are of no concern. An important result that needs to be noted here is the performance of the FFCD model. The FFCD model is relatively simple and is based on the overall results the model has shown to predict different divertor plasma parameters. The FFCD model treats exactly the implementation of the deep penetration of energetic neutrals emerging from the divertor plate. The resulting ionization profiles are
International Nuclear Information System (INIS)
Prinja, A.K.
1998-01-01
In this work, it has been shown that, for the given sets of parameters (transport coefficients), the Tangent-Predictor (TP) continuation method, which was used in the coarsest grid, works remarkably well. The problems in finding an initial guess that resides well within Newton's method radius of convergence are alleviated by correcting the initial guess by the predictor step of the TP method. The TP method works well also in neutral gas puffing and impurity simulations. The neutral gas puffing simulation is performed by systematically increasing the fraction of puffing rate according to the TP method until it reaches a desired condition. Similarly, the impurity simulation characterized by using the fraction of impurity density as the continuation parameter, is carried out in line with the TP method. Both methods show, as expected, a better performance than the classical embedding (CE) method. The convergence criteria ε is set to be 10 -9 based on the fact that lower value of ε does not alter the solution significantly. Correspondingly, the number of Newton's iterations in the corrector step of the TP method decrease substantially, an extra point in terms of code speed. The success of the TP method enlarges the possibility of including other sets of parameters (operations and physics). With the availability of the converged coarsest grid solution, the next forward step to the multigrid cycle becomes possible. The multigrid method shows that the memory storage problems that plagued the application of Newton's method on fine grids, are of no concern. An important result that needs to be noted here is the performance of the FFCD model. The FFCD model is relatively simple and is based on the overall results the model has shown to predict different divertor plasma parameters. The FFCD model treats exactly the implementation of the deep penetration of energetic neutrals emerging from the divertor plate. The resulting ionization profiles are relatively smooth as a
Directory of Open Access Journals (Sweden)
Mojtaba Rasouli Gandomani
2016-06-01
Full Text Available In this paper, the model of HIV infection of CD4+ T cells is considered as a system of fractional differential equations. Then, a numerical method by using collocation method based on the Müntz-Legendre polynomials to approximate solution of the model is presented. The application of the proposed numerical method causes fractional differential equations system to convert into the algebraic equations system. The new system can be solved by one of the existing methods. Finally, we compare the result of this numerical method with the result of the methods have already been presented in the literature.
International Nuclear Information System (INIS)
Menezes, Welton Alves de
2009-01-01
In this dissertation the spectral nodal method SD-SGF-CN, cf. spectral diamond - spectral Green's function - constant nodal, is used to determine the angular fluxes averaged along the edges of the homogenized nodes in heterogeneous domains. Using these results, we developed an algorithm for the reconstruction of the node-edge average angular fluxes within the nodes of the spatial grid set up on the domain, since more localized numerical solutions are not generated by coarse-mesh numerical methods. Numerical results are presented to illustrate the accuracy of the algorithm we offer. (author)
International Nuclear Information System (INIS)
Ixaru, G.L.
1978-03-01
The method developed in the previous paper (preprint, C.I.Ph. (Bucharest), MC-2-78, 1978) is here investigated from computational point of view. Special emphasis is paid to the two basic descriptors of the efficiency: the volume of memory required and the computational effort (timing). Next, two experimental cases are reported. They (i) confirm the theoretical estimates for the rate cf convergence of each version of the present method and (ii) show that the present method is substantially faster than the others. Specifically, it is found that for typical physical problems it is faster by a factor of ten up to twenty than the methods commonly used, viz. Numerov and de Vogelaere. The data reported also allow an inUirect comparison with the method of Gordon. I l/ allow an indirect comparison with the method of Gordon. It is shown that, while this exhibits the same rate as our basic, lowest order version, the computational effort for the latter is, in case of systems with nine equations, only half than for the method of Gordon. At the end of the paper some types of physical problems are suggested which should be the most benefitting if solved numerically with the present method. (author)
Rao, G Shanker
2006-01-01
About the Book: This book provides an introduction to Numerical Analysis for the students of Mathematics and Engineering. The book is designed in accordance with the common core syllabus of Numerical Analysis of Universities of Andhra Pradesh and also the syllabus prescribed in most of the Indian Universities. Salient features: Approximate and Numerical Solutions of Algebraic and Transcendental Equation Interpolation of Functions Numerical Differentiation and Integration and Numerical Solution of Ordinary Differential Equations The last three chapters deal with Curve Fitting, Eigen Values and Eigen Vectors of a Matrix and Regression Analysis. Each chapter is supplemented with a number of worked-out examples as well as number of problems to be solved by the students. This would help in the better understanding of the subject. Contents: Errors Solution of Algebraic and Transcendental Equations Finite Differences Interpolation with Equal Intervals Interpolation with Unequal Int...
Khabaza, I M
1960-01-01
Numerical Analysis is an elementary introduction to numerical analysis, its applications, limitations, and pitfalls. Methods suitable for digital computers are emphasized, but some desk computations are also described. Topics covered range from the use of digital computers in numerical work to errors in computations using desk machines, finite difference methods, and numerical solution of ordinary differential equations. This book is comprised of eight chapters and begins with an overview of the importance of digital computers in numerical analysis, followed by a discussion on errors in comput
Shibata, Masaru
2016-01-01
This book is composed of two parts: First part describes basics in numerical relativity, that is, the formulations and methods for a solution of Einstein's equation and general relativistic matter field equations. This part will be helpful for beginners of numerical relativity who would like to understand the content of numerical relativity and its background. The second part focuses on the application of numerical relativity. A wide variety of scientific numerical results are introduced focusing in particular on the merger of binary neutron stars and black holes.
Energy Technology Data Exchange (ETDEWEB)
Munz, C D; Schneider, R; Stein, E; Voss, U [Forschungszentrum Karlsruhe (Germany). Institut fuer Neutronenphysik und Reaktortechnik; Westermann, T [FH Karlsruhe (Germany). Fachbereich Naturwissenschaften; Krauss, M [Forschungszentrum Karlsruhe (Germany). Hauptabteilung Informations- und Kommunikationstechik
1997-12-31
The numerical concept realized in the the Karlsruhe Diode Code KADI2D is briefly reviewed. Several new aspects concerning the Maxwell field solver based on high resolution finite-volume methods are presented. A new approach maintaining charge conservation numerically for the Maxwell-Lorentz equations is shortly summarized. (author). 2 figs., 12 refs.
International Nuclear Information System (INIS)
Munz, C.D.; Schneider, R.; Stein, E.; Voss, U.; Westermann, T.; Krauss, M.
1996-01-01
The numerical concept realized in the the Karlsruhe Diode Code KADI2D is briefly reviewed. Several new aspects concerning the Maxwell field solver based on high resolution finite-volume methods are presented. A new approach maintaining charge conservation numerically for the Maxwell-Lorentz equations is shortly summarized. (author). 2 figs., 12 refs
Numerical solution of the elastic non-axial contact between pellet and cladding of fuel rod in PWR
International Nuclear Information System (INIS)
Zymak, J.
1987-08-01
Elastic non-axial contacts between the pellet and the cladding of a fuel rod in a pressurized water reactor were calculated. The existence and the uniqueness of the solution were proved. The problem was approximated by the finite element method and quadratic programming was used for the solution. The results will be used in the solution of the probabilistic model of a fuel rod with non-axial pellets in a PWR. (author). 10 figs., 4 tabs., 10 refs
Energy Technology Data Exchange (ETDEWEB)
Weihermueller, L
2005-07-01
To date, the understanding of processes, factors, and interactions that influence the amount of extracted water and the solute composition sampled with suction cups is limited. But this information is required for process description of solute transport in natural soils. Improved system understanding can lead to a low cost and easy to install water sampling system which can help to predict solute transport in natural soils for the benefit of environmental protection. The main objectives of this work were to perform numerical simulations with different boundary conditions and to implement the findings in the interpretation of the lysimeter and field experiments. In a first part of this thesis, theoretical considerations on the processes affecting the spatial influence of a suction cup in soil and changes in solute transport initiated by the suction cups are presented, including testing and validation of available model and experimental approaches. In the second part, a detailed experimental study was conducted to obtain data for the comparison of the different soil water sampling systems. Finally, the numerical experiments of the suction cup influence were used for the interpretation of the experimental data. The main goals are summarized as follows: - Characterization of the suction cup activity domain (SCAD), suction cup extraction domain (SCED) and suction cup sampling area (SCSA) of active suction cups (definitions are given in Chapter 6). - Determination of the boundary conditions and soil properties [e.g. infiltration, applied suction, duration of water extraction, soil hydraulic properties and soil heterogeneity] affecting the activity domain, extraction domain and sampling area of a suction cup. - Identification of processes that change the travel time and travel time variance of solutes extracted by suction cups. - Validation of the numerically derived data with analytical and experimental data from literature. - Comparison of the experimental data obtained
A Well-Posed Two Phase Flow Model and its Numerical Solutions for Reactor Thermal-Fluids Analysis
Energy Technology Data Exchange (ETDEWEB)
Kadioglu, Samet Y. [Idaho National Lab. (INL), Idaho Falls, ID (United States); Berry, Ray [Idaho National Lab. (INL), Idaho Falls, ID (United States); Martineau, Richard [Idaho National Lab. (INL), Idaho Falls, ID (United States)
2016-08-01
A 7-equation two-phase flow model and its numerical implementation is presented for reactor thermal-fluids applications. The equation system is well-posed and treats both phases as compressible flows. The numerical discretization of the equation system is based on the finite element formalism. The numerical algorithm is implemented in the next generation RELAP-7 code (Idaho National Laboratory (INL)’s thermal-fluids code) built on top of an other INL’s product, the massively parallel multi-implicit multi-physics object oriented code environment (MOOSE). Some preliminary thermal-fluids computations are presented.
A Well-Posed Two Phase Flow Model and its Numerical Solutions for Reactor Thermal-Fluids Analysis
International Nuclear Information System (INIS)
Kadioglu, Samet Y.; Berry, Ray; Martineau, Richard
2016-01-01
A 7-equation two-phase flow model and its numerical implementation is presented for reactor thermal-fluids applications. The equation system is well-posed and treats both phases as compressible flows. The numerical discretization of the equation system is based on the finite element formalism. The numerical algorithm is implemented in the next generation RELAP-7 code (Idaho National Laboratory (INL)'s thermal-fluids code) built on top of an other INL's product, the massively parallel multi-implicit multi-physics object oriented code environment (MOOSE). Some preliminary thermal-fluids computations are presented.
He, Qiaolin
2011-06-01
In this article we discuss the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn-Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient. © 2011 Elsevier Inc.
CSIR Research Space (South Africa)
Shatalov, M
2012-09-01
Full Text Available Exact solutions of equations of longitudinal vibration of conical and exponential rod are analyzed for the Rayleigh-Love model. These solutions are used as reference results for checking accuracy of the method of lines. It is shown that the method...
International Nuclear Information System (INIS)
Kumar, Vikas; Gupta, R. K.; Jiwari, Ram
2014-01-01
In this paper, the variable-coefficient diffusion—advection (DA) equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended (G'/G)-expansion method is applied to the DA equation to construct some new non-traveling wave solutions
Energy Technology Data Exchange (ETDEWEB)
Martin Alves, Antonio S. de [NUCLEN, Rio de Janeiro, RJ (Brazil); Passos, Aline M.M. dos [Universidade Federal Fluminense, Niteroi, RJ (Brazil)
1997-12-01
The safety analysis of a structure known as repository for medium activity wastes leads to investigating the physical phenomena connected to the water infiltration. This work shows succinctly an engineering approach to obtain numerical results for the model differential equations. One of these equations, related to the two-phase flow within the structure, is a nonlinear Riccati type, whose solution is only known for certain cases. For safety analysis and design purposes, the solution for the case of variable parameters is also advantageous when one aims some accident scenarios analysis. The utilization of numerical techniques allowed excellent results applied for the design of the Abadia de Goias repository. The case treated in this paper was one of those applied to the safety assessment of this repository. (author). 7 refs., 4 figs., 7 tabs.
Energy Technology Data Exchange (ETDEWEB)
Mingatos, Danielle dos Santos; Bevilacqua, Joyce da Silva, E-mail: dani@ime.usp.br, E-mail: joyce@ime.usp.br [Universidade de Sao Paulo (IME/USP), SP (Brazil). Instituto de Matematica e Estatistica; Todo, Alberto Saburo; Rodrigues Junior, Orlando, E-mail: astodo@ipen.br, E-mail: rodrijr@ipen.br [Instituto de Pesquisas Energeticas Nucleares (IPEN/CNEN-SP), Sao Paulo, SP (Brazil)
2013-07-01
Biokinetics models for radionuclides applied to dosimetry problems are constantly reviewed by ICRP. The radionuclide trajectory could be represented by compartmental models, assuming constant transfer rates between compartments. A better understanding of physiological or biochemical phenomena, improve the comprehension of radionuclide behavior in the human body and, in general, more complex compartmental models are proposed, increasing the difficulty of obtaining the analytical solution for the system of first order differential equations. Even with constant transfer rates numerical solutions must be carefully implemented because of almost singular characteristic of the matrix of coefficients. In this work we compare numerical methods with different strategies for ICRP-78 models for Thorium-228 and Uranium-234. The impact of uncertainty in the parameters of the equations is also estimated for local and global truncation errors. (author)
International Nuclear Information System (INIS)
Aguilera, V.M.; Garrido, J.; Mafe, S.; Pellicer, J.
1985-01-01
An algorithm for the solution of Nernst-Planck equations with simultaneous convective flux and electric current has been developed without using Poisson's equation. The numerical simulation which has been developed reproduces the behaviour of the system employing their experimental variables as parameters of the algorithm. However, other procedures are only capable of dealing with some of the experimental conditions described here. The agreement between the theoretically predicted values and the experimentally obtained is quite reasonable. (author)
Chavez Chavez, Gustavo Ivan
2017-01-01
Numerical experiments corroborate the robustness, accuracy, and complexity claims and provide a baseline of the performance and memory footprint by comparisons with competing approaches such as the multigrid solver hypre, and the STRUMPACK implementation of the multifrontal factorization with hierarchically semi-separable matrices. The companion implementation can utilize many thousands of cores of Shaheen, KAUST's Haswell-based Cray XC-40 supercomputer, and compares favorably with other implementations of hierarchical solvers in terms of time-to-solution and memory consumption.
1982-04-01
would have difficulty resolving regions of high curvature (for example the local maxima of sharp peaks in the solution). Most of the ideas discussed above...use of improved methods and implementations as they become available. We anticipate that by using the package the sofware development time and...the precise control of the grid at interior points), solution adaptive methods to better resolve the maxima in the E-fields, and semi-automated
Hozman, J.; Tichý, T.
2017-12-01
Stochastic volatility models enable to capture the real world features of the options better than the classical Black-Scholes treatment. Here we focus on pricing of European-style options under the Stein-Stein stochastic volatility model when the option value depends on the time, on the price of the underlying asset and on the volatility as a function of a mean reverting Orstein-Uhlenbeck process. A standard mathematical approach to this model leads to the non-stationary second-order degenerate partial differential equation of two spatial variables completed by the system of boundary and terminal conditions. In order to improve the numerical valuation process for a such pricing equation, we propose a numerical technique based on the discontinuous Galerkin method and the Crank-Nicolson scheme. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on options with stochastic volatility.
On full-tensor permeabilities of porous media from numerical solutions of the Navier-Stokes equation
Wang, Y.; Sun, S.; Yu, B.
2013-01-01
A numerical method is proposed to compute full-tensor permeability of porous media without artificial simplification. Navier-Stokes (N-S) equation and Darcy's law are combined to design these numerical experiments. This method can successfully detect the permeability values in principle directions of the porous media and the anisotropic degrees. It is found that the same configuration of porous media may possess isotropic features at lower Reynolds numbers while manifesting anisotropic features at higher Reynolds numbers due to the nonlinearity from convection. Anisotropy becomes pronounced especially when convection is dominant. 2013 Yi Wang et al.
Czech Academy of Sciences Publication Activity Database
Feireisl, Eduard; Medviďová-Lukáčová, M.; Nečasová, Šárka; Novotný, A.; She, Bangwei
2018-01-01
Roč. 16, č. 1 (2018), s. 150-183 ISSN 1540-3459 R&D Projects: GA ČR GA16-03230S EU Projects: European Commission(XE) 320078 - MATHEF Institutional support: RVO:67985840 Keywords : Navier-Stokes system * finite element numerical method * finite volume numerical method * asymptotic preserving schemes Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.865, year: 2016 http://epubs.siam.org/doi/10.1137/16M1094233
On full-tensor permeabilities of porous media from numerical solutions of the Navier-Stokes equation
Wang, Y.
2013-01-01
A numerical method is proposed to compute full-tensor permeability of porous media without artificial simplification. Navier-Stokes (N-S) equation and Darcy\\'s law are combined to design these numerical experiments. This method can successfully detect the permeability values in principle directions of the porous media and the anisotropic degrees. It is found that the same configuration of porous media may possess isotropic features at lower Reynolds numbers while manifesting anisotropic features at higher Reynolds numbers due to the nonlinearity from convection. Anisotropy becomes pronounced especially when convection is dominant. 2013 Yi Wang et al.
Zhang, Kejiang; Achari, Gopal; Li, Hua
2009-11-03
Traditionally, uncertainty in parameters are represented as probabilistic distributions and incorporated into groundwater flow and contaminant transport models. With the advent of newer uncertainty theories, it is now understood that stochastic methods cannot properly represent non random uncertainties. In the groundwater flow and contaminant transport equations, uncertainty in some parameters may be random, whereas those of others may be non random. The objective of this paper is to develop a fuzzy-stochastic partial differential equation (FSPDE) model to simulate conditions where both random and non random uncertainties are involved in groundwater flow and solute transport. Three potential solution techniques namely, (a) transforming a probability distribution to a possibility distribution (Method I) then a FSPDE becomes a fuzzy partial differential equation (FPDE), (b) transforming a possibility distribution to a probability distribution (Method II) and then a FSPDE becomes a stochastic partial differential equation (SPDE), and (c) the combination of Monte Carlo methods and FPDE solution techniques (Method III) are proposed and compared. The effects of these three methods on the predictive results are investigated by using two case studies. The results show that the predictions obtained from Method II is a specific case of that got from Method I. When an exact probabilistic result is needed, Method II is suggested. As the loss or gain of information during a probability-possibility (or vice versa) transformation cannot be quantified, their influences on the predictive results is not known. Thus, Method III should probably be preferred for risk assessments.
Zhuravlev, A. K.; Anokhin, A. O.; Irkhin, V. Yu.
2018-02-01
Simple scaling consideration and NRG solution of the one- and two-channel Kondo model in the presence of a logarithmic Van Hove singularity at the Fermi level is given. The temperature dependences of local and impurity magnetic susceptibility and impurity entropy are calculated. The low-temperature behavior of the impurity susceptibility and impurity entropy turns out to be non-universal in the Kondo sense and independent of the s-d coupling J. The resonant level model solution in the strong coupling regime confirms the NRG results. In the two-channel case the local susceptibility demonstrates a non-Fermi-liquid power-law behavior.
Energy Technology Data Exchange (ETDEWEB)
Molinero, J.; Samper, J.
2003-07-01
Several countries around the world are considering the final disposal of high-level radioactive waste in deep repositories located in fractured granite formations. Evaluating the long term safety of such repositories requires sound conceptual and numerical models which must consider simultaneously groundwater flow, solute transport and chemical and radiological processes. These models are being developed from data and knowledge gained from in situ experiments carried out at deep underground laboratories such as that of Aspo, Sweden, constructed in fractured granite. The Redox Zone Experiment is one of such experiments performed at Aspo in order to evaluate the effects of the construction of the access tunnel on the hydrogeological and hydrochemical conditions of a fracture zone intersected by the tunnel. Previous authors interpreted hydrochemical and isotopic data of this experiment using a mass-balance approach based on a qualitative description of groundwater flow conditions. Such an interpretation, however, is subject to uncertainties related to an over-simplified conceptualization of groundwater flow. Here we present numerical models of groundwater flow and solute transport for this fracture zone. The first model is based on previously published conceptual model. It presents noticeable un consistencies and fails to match simultaneously observed draw downs and chloride breakthrough curves. To overcome its limitations, a revised flow and transport model is presented which relies directly on available hydrodynamic and transport parameters, is based on the identification of appropriate flow and transport boundary conditions and uses, when needed, solute data extrapolated from nearby fracture zones. A significant quantitative improvement is achieved with the revised model because its results match simultaneously drawdown and chloride data. Other improvements are qualitative and include: ensuring consistency of hydrodynamic and hydrochemical data and avoiding
Energy Technology Data Exchange (ETDEWEB)
Samaras, T; Christ, A; Kuster, N [Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki (Greece); Foundation for Research on Information Technologies in Society (IT' IS Foundation), Swiss Federal Institute of Technology (ETH), CH-8004 Zurich (Switzerland)
2006-06-07
In this work, we highlight two issues that have to be taken into consideration for accurate thermal modelling with the finite-difference time-domain (FDTD) method, namely the tissue interfaces and the staircasing effect. The former appears less critical in the overall accuracy of the results, whereas the latter may have an influence on the worst-case approach used in numerical dosimetry of non-ionizing radiation. (note)
International Nuclear Information System (INIS)
Samaras, T; Christ, A; Kuster, N
2006-01-01
In this work, we highlight two issues that have to be taken into consideration for accurate thermal modelling with the finite-difference time-domain (FDTD) method, namely the tissue interfaces and the staircasing effect. The former appears less critical in the overall accuracy of the results, whereas the latter may have an influence on the worst-case approach used in numerical dosimetry of non-ionizing radiation. (note)
Directory of Open Access Journals (Sweden)
M.J. Uddin
2016-09-01
Full Text Available The two-dimensional unsteady laminar free convective heat and mass transfer fluid flow of a non-Newtonian fluid adjacent to a vertical plate has been analyzed numerically. The two parameters Lie group transformation method that transforms the three independent variables into a single variable is used to transform the continuity, the momentum, the energy and the concentration equations into a set of coupled similarity equations. The transformed equations have been solved by the Runge–Kutta–Fehlberg fourth-fifth order numerical method with shooting technique. Numerical calculations were carried out for the various parameters entering into the problem. The dimensionless velocity, temperature and concentration profiles were shown graphically and the skin friction, heat and mass transfer rates were given in tables. It is found that friction factor and heat transfer (mass transfer rate for methanol are higher (lower than those of hydrogen and water vapor. Friction factor decreases while heat and mass transfer rate increase as the Prandtl number increases. Friction (heat and mass transfer rate factor of Newtonian fluid is higher (lower than the dilatant fluid.
Energy Technology Data Exchange (ETDEWEB)
Paster, Amir, E-mail: paster@tau.ac.il [Environmental Fluid Mechanics Laboratories, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN (United States); School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978 (Israel); Bolster, Diogo [Environmental Fluid Mechanics Laboratories, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN (United States); Benson, David A. [Hydrologic Science and Engineering, Colorado School of Mines, Golden, CO, 80401 (United States)
2014-04-15
We study a system with bimolecular irreversible kinetic reaction A+B→∅ where the underlying transport of reactants is governed by diffusion, and the local reaction term is given by the law of mass action. We consider the case where the initial concentrations are given in terms of an average and a white noise perturbation. Our goal is to solve the diffusion–reaction equation which governs the system, and we tackle it with both analytical and numerical approaches. To obtain an analytical solution, we develop the equations of moments and solve them approximately. To obtain a numerical solution, we develop a grid-less Monte Carlo particle tracking approach, where diffusion is modeled by a random walk of the particles, and reaction is modeled by annihilation of particles. The probability of annihilation is derived analytically from the particles' co-location probability. We rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number. This enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies. - Graphical abstract:.
Simos, T. E.
2017-11-01
A family of four stages high algebraic order embedded explicit six-step methods, for the numerical solution of second order initial or boundary-value problems with periodical and/or oscillating solutions, are studied in this paper. The free parameters of the new proposed methods are calculated solving the linear system of equations which is produced by requesting the vanishing of the phase-lag of the methods and the vanishing of the phase-lag's derivatives of the schemes. For the new obtained methods we investigate: • Its local truncation error (LTE) of the methods.• The asymptotic form of the LTE obtained using as model problem the radial Schrödinger equation.• The comparison of the asymptotic forms of LTEs for several methods of the same family. This comparison leads to conclusions on the efficiency of each method of the family.• The stability and the interval of periodicity of the obtained methods of the new family of embedded finite difference pairs.• The applications of the new obtained family of embedded finite difference pairs to the numerical solution of several second order problems like the radial Schrödinger equation, astronomical problems etc. The above applications lead to conclusion on the efficiency of the methods of the new family of embedded finite difference pairs.
International Nuclear Information System (INIS)
Baptista Filho, Benedito Dias
1979-01-01
A numerical model has been developed to calculate the flow, pressure and temperature distribution of steady-state |for the tube and shell-side fluids in a shell-and-U-tubes heat exchanger with segmental baffles. It was based on the Subchannel Analysis Method- The model, checked with experimental results from one heat exchanger, predicted with good accuracy outlet temperatures for both fluids. The method, implemented ' in a computer program of low cost and easy application, can be used in the design and performance evaluation of commercial units.(author)
International Nuclear Information System (INIS)
Mahaffy, J.H.; Liles, D.R.
1977-01-01
A numerical method for treating two-phase flow in pipes is presented which incorporates the use of a partially implicit scheme in regions of relatively low flow velocity and a fully implicit treatment in regions of high velocity. This method takes advantage of the lower cost per iteration of the partially implicit scheme, without being limited by its conditional stability. Applications of this approach to water reactor blowdown calculations produce reductions in computer time by factors of 2 to 4 without a significant loss of accuracy