Most of thermal hydraulic processes in nuclear engineering can be described by general convection-diffusion equations that are often can be simulated numerically with finite-difference method (FDM). An effective scheme for finite-difference discretization of such equations is presented in this report. The derivation of this scheme is based on analytical solutions of a simplified one-dimensional equation written for every control volume of the finite-difference mesh. These analytical solutions are constructed using linearized representations of both diffusion coefficient and source term. As a result, the Efficient Finite-Differencing (EFD) scheme makes it possible to significantly improve the accuracy of numerical method even using mesh systems with fewer grid nodes that, in turn, allows to speed-up numerical simulation. EFD has been carefully verified on the series of sample problems for which either analytical or very precise numerical solutions can be found. EFD has been compared with other popular FDM schemes including novel, accurate (as well as sophisticated) methods. Among the methods compared were well-known central difference scheme, upwind scheme, exponential differencing and hybrid schemes of Spalding. Also, newly developed finite-difference schemes, such as the the quadratic upstream (QUICK) scheme of Leonard, the locally analytic differencing (LOAD) scheme of Wong and Raithby, the flux-spline scheme proposed by Varejago and Patankar as well as the latest LENS discretization of Sakai have been compared. Detailed results of this comparison are given in this report. These tests have shown a high efficiency of the EFD scheme. For most of sample problems considered EFD has demonstrated the numerical error that appeared to be in orders of magnitude lower than that of other discretization methods. Or, in other words, EFD has predicted numerical solution with the same given numerical error but using much fewer grid nodes. In this report, the detailed
Nonstandard finite difference schemes
Mickens, Ronald E.
1995-01-01
The major research activities of this proposal center on the construction and analysis of nonstandard finite-difference schemes for ordinary and partial differential equations. In particular, we investigate schemes that either have zero truncation errors (exact schemes) or possess other significant features of importance for numerical integration. Our eventual goal is to bring these methods to bear on problems that arise in the modeling of various physical, engineering, and technological systems. At present, these efforts are extended in the direction of understanding the exact nature of these nonstandard procedures and extending their use to more complicated model equations. Our presentation will give a listing (obtained to date) of the nonstandard rules, their application to a number of linear and nonlinear, ordinary and partial differential equations. In certain cases, numerical results will be presented.
On stochastic finite difference schemes
Gyongy, Istvan
2013-01-01
Finite difference schemes in the spatial variable for degenerate stochastic parabolic PDEs are investigated. Sharp results on the rate of $L_p$ and almost sure convergence of the finite difference approximations are presented and results on Richardson extrapolation are established for stochastic parabolic schemes under smoothness assumptions.
Lei Zhang; Lisha Wang; Xiaohua Ding
2014-01-01
We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.
Implicit finite difference schemes for the magnetic induction equations
Koley, U.
2011-01-01
We describe high order accurate and stable fully-discrete finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that d...
A Finite Element Framework for Some Mimetic Finite Difference Discretizations
Rodrigo, Carmen; Gaspar, Francisco; Hu, Xiaozhe; Zikatanov, Ludmil
2015-01-01
In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\\'ed\\'elec-Raviart-Thomas finite element methods for model problems in $\\mathbf{H}(\\operatorname{\\mathbf{curl}})$ and $H(\\operatorname{div})$. This provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well-known results from finite element theory. The finite element framework that we develop is also crucial for the design...
Efficient discretization in finite difference method
Rozos, Evangelos; Koussis, Antonis; Koutsoyiannis, Demetris
2015-04-01
Finite difference method (FDM) is a plausible and simple method for solving partial differential equations. The standard practice is to use an orthogonal discretization to form algebraic approximate formulations of the derivatives of the unknown function and a grid, much like raster maps, to represent the properties of the function domain. For example, for the solution of the groundwater flow equation, a raster map is required for the characterization of the discretization cells (flow cell, no-flow cell, boundary cell, etc.), and two raster maps are required for the hydraulic conductivity and the storage coefficient. Unfortunately, this simple approach to describe the topology comes along with the known disadvantages of the FDM (rough representation of the geometry of the boundaries, wasted computational resources in the unavoidable expansion of the grid refinement in all cells of the same column and row, etc.). To overcome these disadvantages, Hunt has suggested an alternative approach to describe the topology, the use of an array of neighbours. This limits the need for discretization nodes only for the representation of the boundary conditions and the flow domain. Furthermore, the geometry of the boundaries is described more accurately using a vector representation. Most importantly, graded meshes can be employed, which are capable of restricting grid refinement only in the areas of interest (e.g. regions where hydraulic head varies rapidly, locations of pumping wells, etc.). In this study, we test the Hunt approach against MODFLOW, a well established finite difference model, and the Finite Volume Method with Simplified Integration (FVMSI). The results of this comparison are examined and critically discussed.
Applications of nonstandard finite difference schemes
Mickens, Ronald E
2000-01-01
The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter 1 gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain
Vibration source identification using corrected finite difference schemes
Leclere, Q.; Pezerat, Charles
2012-01-01
International audience This paper addresses the problem of the location and identification of vibration excitations from the measurement of the displacement field of a vibrating structure. It constitutes an improvement of the force analysis technique published several years ago. The development is based on the use of the motion equation which is discretized by finite difference schemes approximating spatial derivatives of the displacement. In a first instance, the error due to this approxi...
An optimized finite-difference scheme for wave propagation problems
Zingg, D. W.; Lomax, H.; Jurgens, H.
1993-01-01
Two fully-discrete finite-difference schemes for wave propagation problems are presented, a maximum-order scheme and an optimized (or spectral-like) scheme. Both combine a seven-point spatial operator and an explicit six-stage time-march method. The maximum-order operator is fifth-order in space and is sixth-order in time for a linear problem with periodic boundary conditions. The phase and amplitude errors of the schemes obtained using Fourier analysis are given and compared with a second-order and a fourth-order method. Numerical experiments are presented which demonstrate the usefulness of the schemes for a range of problems. For some problems, the optimized scheme leads to a reduction in global error compared to the maximum-order scheme with no additional computational expense.
Finite-difference schemes for anisotropic diffusion
Es, Bram van, E-mail: es@cwi.nl [Centrum Wiskunde and Informatica, P.O. Box 94079, 1090GB Amsterdam (Netherlands); FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM (Netherlands); Koren, Barry [Eindhoven University of Technology (Netherlands); Blank, Hugo J. de [FOM Institute DIFFER, Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM (Netherlands)
2014-09-01
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.
Lie group computation of finite difference schemes
Hoarau, Emma; David, Claire
2006-01-01
nombre de pages 10 A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation, by means of its differential representation. The package provides functions which enable us to solve the determining equations of the related Lie group
Optimizations on Designing High-Resolution Finite-Difference Schemes
Liu, Yen; Koomullil, George; Kwak, Dochan (Technical Monitor)
1994-01-01
We describe a general optimization procedure for both maximizing the resolution characteristics of existing finite differencing schemes as well as designing finite difference schemes that will meet the error tolerance requirements of numerical solutions. The procedure is based on an optimization process. This is a generalization of the compact scheme introduced by Lele in which the resolution is improved for single, one-dimensional spatial derivative, whereas in the present approach the complete scheme, after spatial and temporal discretizations, is optimized on a range of parameters of the scheme and the governing equations. The approach is to linearize and Fourier analyze the discretized equations to check the resolving power of the scheme for various wave number ranges in the solution and optimize the resolution to satisfy the requirements of the problem. This represents a constrained nonlinear optimization problem which can be solved to obtain the nodal weights of discretization. An objective function is defined in the parametric space of wave numbers, Courant number, Mach number and other quantities of interest. Typical criterion for defining the objective function include the maximization of the resolution of high wave numbers for acoustic and electromagnetic wave propagations and turbulence calculations. The procedure is being tested on off-design conditions of non-uniform mesh, non-periodic boundary conditions, and non-constant wave speeds for scalar and system of equations. This includes the solution of wave equations and Euler equations using a conventional scheme with and without optimization and the design of an optimum scheme for the specified error tolerance.
Ferreira, J. A.
2004-01-01
In this paper we study the convergence properties of a finite difference discretization of a second order elliptic equation with mixed derivatives and variable coefficient in polygonal domains subject to general boundary conditions. We prove that the finite difference scheme on nonuniform grids exhibit the phenomenon of supraconvergence, more precisely, for s ∈ [1, 2] order O(hs)-convergence of the finite difference solution and its gradient if the exact solution is in the Sobo...
A New Class of Finite Difference Schemes
Mahesh, K.
1996-01-01
Fluid flows in the transitional and turbulent regimes possess a wide range of length and time scales. The numerical computation of these flows therefore requires numerical methods that can accurately represent the entire, or at least a significant portion, of this range of scales. The inaccurate representation of small scales is inherent to non-spectral schemes. This can be detrimental to computations where the energy in the small scales is comparable to that in the larger scales, e.g. large-eddy simulations of high Reynolds number turbulence. The inaccurate numerical representation of the small scales in these large-eddy simulations can result in the numerical error overwhelming the contribution of the subgrid-scale model.
Higher order finite difference schemes for the magnetic induction equations
Koley, Ujjwal; Mishra, Siddhartha; Risebro, Nils Henrik; Svärd, Magnus
2011-01-01
We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both...
Finite difference schemes for long-time integration
Haras, Zigo; Taasan, Shlomo
1993-01-01
Finite difference schemes for the evaluation of first and second derivatives are presented. These second order compact schemes were designed for long-time integration of evolution equations by solving a quadratic constrained minimization problem. The quadratic cost function measures the global truncation error while taking into account the initial data. The resulting schemes are applicable for integration times fourfold, or more, longer than similar previously studied schemes. A similar approach was used to obtain improved integration schemes.
Scheme For Finite-Difference Computations Of Waves
Davis, Sanford
1992-01-01
Compact algorithms generating and solving finite-difference approximations of partial differential equations for propagation of waves obtained by new method. Based on concept of discrete dispersion relation. Used in wave propagation to relate frequency to wavelength and is key measure of wave fidelity.
Stability of central finite difference schemes for the Heston PDE
Hout, K. J. in 't; K. Volders
2010-01-01
This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments.
Compact finite difference schemes with spectral-like resolution
Lele, Sanjiva K.
1992-01-01
The present finite-difference schemes for the evaluation of first-order, second-order, and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes. Various boundary conditions may be invoked, and both accurate interpolation and spectral-like filtering can be accomplished by means of schemes for derivatives at mid-cell locations. This family of schemes reduces to the Pade schemes when the maximal formal accuracy constraint is imposed with a specific computational stencil. Attention is given to illustrative applications of these schemes in fluid dynamics.
Stability analysis of a finite difference scheme using symbolic computation
Computer aided symbolic manipulation has previously been used in studies of weakly nonlinear behavior in plasmas. The application of such symbolic manipulation techniques to the study of the stability of a specific finite difference scheme is investigated. Eigenvalue calculations for the Von Neumann matrix are described
ADI Finite Difference Discretization of the Heston-Hull-White PDE
Haentjens, Tinne; Hout, Karel in't.
2010-09-01
This paper concerns the efficient numerical solution of the time-dependent, three-dimensional Heston-Hull-White PDE for the fair prices of European call options. The numerical solution method described in this paper consists of a finite difference discretization on non-uniform spatial grids followed by an Alternating Direction Implicit scheme for the time discretization and extends the method recently proved effective by In't Hout & Foulon (2010) for the simpler, two-dimensional Heston PDE.
Ewing, R.; Iliev, O.; Lazarov, R.; Naumovich, A.
2004-01-01
Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of conve...
Convergence of finite difference schemes to the Aleksandrov solution of the Monge-Ampere equation
Awanou, Gerard; Awi, Romeo
2015-01-01
We present a technique for proving convergence to the Aleksandrov solution of the Monge-Ampere equation of a stable and consistent finite difference scheme. We also require a notion of discrete convexity with a stability property and a local equicontinuity property for bounded sequences.
To the convergence ov finite difference schemes on the generalized solutions of the Poisson equation
The rate of convergence of the finite difference schemes on the generalized solutions of Poisson equation is studied by the energy inequality method. The truncation error is analyzed using Bramle-Hilbert lemma. It is proved that commonly used life-point difference scheme for the Direchlet boundary value problem converges with the rate of O(hsup(1+s)) in a discrete Ws21 norm, and with the rate of O(hsup(1+s)|lnh|sup(1/2)) in discrete C-norm, if the solution is from Wsub(2)sup(2+s), s=0.1
Discretizing delta functions via finite differences and gradient normalization
Towers, John D.
2009-06-01
In [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915-931] the author presented two closely related finite difference methods (referred to here as FDM1 and FDM2) for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u :Rn ↦ R . These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h → 0) in the codimension one setting. In this paper, we concentrate on n ⩽ 3 , but generalize our methods to codimensions other than one - now the level set function is generally a vector valued mapping u → :Rn ↦Rm, 1 ⩽ m ⩽ n ⩽ 3 . Seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems. Motivated by this, we instead use the wedge product formalism to generalize our FDM algorithms, and this approach results in accurate, often consistent approximations. With the goal of ensuring consistency in general, we propose a new gradient normalization process that is applied before our FDM algorithms. These combined algorithms seem to be consistent in all reasonable situations, with numerical experiments indicating O (h2) convergence for our new gradient-normalized FDM2 algorithm. In the full codimension setting (m = n) , our gradient normalization processing also improves accuracy when using more standard approximate delta functions. This combination also yields approximations that appear to be consistent.
Christlieb, Andrew J.; Liu, Yuan; Tang, Qi; Xu, Zhengfu
2014-01-01
In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition and positivity of...
Hong, Qianying; Lai, Ming-Jun; Wang, Jingyue
2013-01-01
We present a convergence analysis of a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fatemi model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme. An application for image denoising is given.
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
Li, Xiao; Qiao, ZhongHua; Zhang, Hui
2016-09-01
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.
Nikkar, Samira; Nordström, Jan
2015-06-01
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
Converged accelerated finite difference scheme for the multigroup neutron diffusion equation
Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration. (authors)
A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations
Gerritsen, Margot; Olsson, Pelle
1996-01-01
We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.
High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains
Fisher, Travis C.; Carpenter, Mark H.
2013-01-01
Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms.
Finite-difference scheme for the numerical solution of the Schroedinger equation
Mickens, Ronald E.; Ramadhani, Issa
1992-01-01
A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.
A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model
Jian Huang; Zhongdi Cen; Anbo Le
2013-01-01
We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodiffere...
Higher order finite difference schemes for the magnetic induction equations with resistivity
Koley, U.; S Mishra; Risebro, N. H.; Svard, And M.
2011-01-01
In this paper, we design high order accurate and stable finite difference schemes for the initial-boundary value problem, associated with the magnetic induction equation with resistivity. We use Summation-By-Parts (SBP) finite difference operators to approximate spatial derivatives and a Simultaneous Approximation Term (SAT) technique for implementing boundary conditions. The resulting schemes are shown to be energy stable. Various numerical experiments demonstrating both the stability and th...
A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation
Akbar Mohebbi
2013-01-01
In this paper, a high-order and unconditionally stable difference method is proposed for the numerical solution of one-space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative of this equation and a Pade approximation of fifth-order for the resulting system of ordinary differential equations. It is shown through analysis that the proposed scheme is unconditionally stable. This new method is easy to imp...
Accuracy of spectral and finite difference schemes in 2D advection problems
Naulin, V.; Nielsen, A.H.
2003-01-01
In this paper we investigate the accuracy of two numerical procedures commonly used to solve 2D advection problems: spectral and finite difference (FD) schemes. These schemes are widely used, simulating, e.g., neutral and plasma flows. FD schemes have long been considered fast, relatively easy to...
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromagnetic properties of the model are symmetric with respect ...
Development and application of a third order scheme of finite differences centered in mesh
In this work the development of a third order scheme of finite differences centered in mesh is presented and it is applied in the numerical solution of those diffusion equations in multi groups in stationary state and X Y geometry. Originally this scheme was developed by Hennart and del Valle for the monoenergetic diffusion equation with a well-known source and they show that the one scheme is of third order when comparing the numerical solution with the analytical solution of a model problem using several mesh refinements and boundary conditions. The scheme by them developed it also introduces the application of numeric quadratures to evaluate the rigidity matrices and of mass that its appear when making use of the finite elements method of Galerkin. One of the used quadratures is the open quadrature of 4 points, no-standard, of Newton-Cotes to evaluate in approximate form the elements of the rigidity matrices. The other quadrature is that of 3 points of Radau that it is used to evaluate the elements of all the mass matrices. One of the objectives of these quadratures are to eliminate the couplings among the Legendre moments 0 and 1 associated to the left and right faces as those associated to the inferior and superior faces of each cell of the discretization. The other objective is to satisfy the particles balance in weighed form in each cell. In this work it expands such development to multiplicative means considering several energy groups. There are described diverse details inherent to the technique, particularly those that refer to the simplification of the algebraic systems that appear due to the space discretization. Numerical results for several test problems are presented and are compared with those obtained with other nodal techniques. (Author)
Nonstandard finite difference scheme for SIRS epidemic model with disease-related death
Fitriah, Z.; Suryanto, A.
2016-04-01
It is well known that SIRS epidemic with disease-related death can be described by a system of nonlinear ordinary differential equations (NL ODEs). This model has two equilibrium points where their existence and stability properties are determined by the basic reproduction number [1]. Besides the qualitative properties, it is also often needed to solve the system of NL ODEs. Euler method and 4th order Runge-Kutta (RK4) method are often used to solve the system of NL ODEs. However, both methods may produce inconsistent qualitative properties of the NL ODEs such as converging to wrong equilibrium point, etc. In this paper we apply non-standard finite difference (NSFD) scheme (see [2,3]) to approximate the solution of SIRS epidemic model with disease-related death. It is shown that the discrete system obtained by NSFD scheme is dynamically consistent with the continuous model. By our numerical simulations, we find that the solutions of NSFD scheme are always positive, bounded and convergent to the correct equilibrium point for any step size of integration (h), while those of Euler or RK4 method have the same properties only for relatively small h.
Stability of finite difference schemes for generalized von Foerster equations with renewal
Henryk Leszczyński
2014-01-01
Full Text Available We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \\(l^1\\ and \\(l^\\infty\\ norms.
Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme
Srivastava, Vineet K.; Sarita Singh; Mukesh K. Awasthi
2013-01-01
In this paper, an implicit exponential finite-difference scheme (Expo FDM) has been proposed for solving two dimensional nonlinear coupled viscous Burgers’ equations (VBEs) with appropriate initial and boundary conditions. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with exact solution and those already available in the literature by finding the L1, L2, L∞ and ER errors. Excellent numerical results indicate that the proposed scheme is...
Nonlinear Comparison of High-Order and Optimized Finite-Difference Schemes
Hixon, R.
1998-01-01
The effect of reducing the formal order of accuracy of a finite-difference scheme in order to optimize its high-frequency performance is investigated using the I-D nonlinear unsteady inviscid Burgers'equation. It is found that the benefits of optimization do carry over into nonlinear applications. Both explicit and compact schemes are compared to Tam and Webb's explicit 7-point Dispersion Relation Preserving scheme as well as a Spectral-like compact scheme derived following Lele's work. Results are given for the absolute and L2 errors as a function of time.
Li, Y.; Han, B.; Métivier, L.; Brossier, R.
2016-09-01
We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg-Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poisson's ratios. Moreover, only 3.7 grid-points per minimum shear wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates wavefields having smaller error under the same discretization setups. Profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.
High Order Finite Difference Schemes for the Elastic Wave Equation in Discontinuous Media
Virta, Kristoffer
2013-01-01
Finite difference schemes for the simulation of elastic waves in materi- als with jump discontinuities are presented. The key feature is the highly accurate treatment of interfaces where media discontinuities arise. The schemes are constructed using finite difference operators satisfying a sum- mation - by - parts property together with a penalty technique to impose interface conditions at the material discontinuity. Two types of opera- tors are used, termed fully compatible or compatible. Stability is proved for the first case by bounding the numerical solution by initial data in a suitably constructed semi - norm. Numerical experiments indicate that the schemes using compatible operators are also stable. However, the nu- merical studies suggests that fully compatible operators give identical or better convergence and accuracy properties. The numerical experiments are also constructed to illustrate the usefulness of the proposed method to simulations involving typical interface phenomena in elastic materials...
A multigrid algorithm for the cell-centered finite difference scheme
Ewing, Richard E.; Shen, Jian
1993-01-01
In this article, we discuss a non-variational V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme may fail to be a contraction. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time with very promising computational results.
A new finite difference scheme for a dissipative cubic nonlinear Schrödinger equation
This paper considers the one-dimensional dissipative cubic nonlinear Schrödinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient. (general)
Fuhrmann, David R.; Bingham, Harry B.; Madsen, Per A.;
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly nonlinear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...... rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water nonlinearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only...... moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local nonlinear analysis. The various methods of analysis combine to provide significant...
A fourth order accurate finite difference scheme for the computation of elastic waves
Bayliss, A.; Jordan, K. E.; Lemesurier, B. J.; Turkel, E.
1986-01-01
A finite difference for elastic waves is introduced. The model is based on the first order system of equations for the velocities and stresses. The differencing is fourth order accurate on the spatial derivatives and second order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here it is found that the fourth order scheme requires for two-thirds to one-half the resolution of a typical second order scheme to give comparable accuracy.
Fisher, Travis C.; Carpenter, Mark H.; Yamaleev, Nail K.; Frankel, Steven H.
2009-01-01
A general strategy exists for constructing Energy Stable Weighted Essentially Non Oscillatory (ESWENO) finite difference schemes up to eighth-order on periodic domains. These ESWENO schemes satisfy an energy norm stability proof for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain wherever possible the WENO stencil biasing properties, while satisfying the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L2 norm. Second-order, and third-order boundary closures are developed that achieve stability in diagonal and block norms, respectively. The global accuracy for the second-order closures is three, and for the third-order closures is four. A novel set of non-uniform flux interpolation points is necessary near the boundaries to simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencil biasing mechanics.
Landing gear noise prediction using high-order finite difference schemes
Liu, W.; Kim, J. W.; Zhang, X; Angland, D.; BASTIEN, C
2012-01-01
Aerodynamic noise from a generic two-wheel landing-gear model is predicted by a CFD/FW-H hybrid approach. The unsteady flow-field is computed using a compressible Navier–Stokes solver based on high-order finite difference schemes and a fully structured grid. The calculated time history of the surface pressure data is used in an FW-H solver to predict the far-field noise levels. Both aerodynamic and aeroacoustic results are compared to wind tunnel measurements and are found to be in good agree...
A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws
Zhu, Jun; Qiu, Jianxian
2016-08-01
In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme [14,20], could get less absolute truncation errors in L1 and L∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme [14,20] in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.
Computational Aero-Acoustic Using High-order Finite-Difference Schemes
Zhu, Wei Jun; Shen, Wen Zhong; Sørensen, Jens Nørkær
2007-01-01
In this paper, a high-order technique to accurately predict flow-generated noise is introduced. The technique consists of solving the viscous incompressible flow equations and inviscid acoustic equations using a incompressible/compressible splitting technique. The incompressible flow equations are...... discretizations of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization....
A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation
Akbar Mohebbi
2013-10-01
Full Text Available In this paper, a high-order and unconditionally stable difference method is proposed for the numerical solution of one-space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative of this equation and a Pade approximation of fifth-order for the resulting system of ordinary differential equations. It is shown through analysis that the proposed scheme is unconditionally stable. This new method is easy to implement, produces very accurate results and needs short CPU time. Some numerical examples are included to demonstrate the validity and applicability of the technique. We compare the numerical results of this paper with the numerical results of some methods in the literature.
Augusta, Petr; Cichy, B.; Galkowski, K.; Rogers, E.
Vila Real: IEEE, 2015, s. 134-139. ISBN 978-1-4799-8739-9. [The 2015 IEEE 9th International Workshop on MultiDimensional (nD) Systems (nDS) (2015). Vila Real (PT), 09.09.2015-11.09.2015] Institutional support: RVO:67985556 Keywords : Discretization * implicit difference scheme * repetitive processes Subject RIV: BC - Control Systems Theory
Chu, Kevin T.
2008-01-01
In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique is its ability to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward numerical analysis a...
Bhattacharya, Amitabh
2013-11-01
An efficient algorithm for simulating Stokes flow around particles is presented here, in which a second order Finite Difference method (FDM) is coupled to a Boundary Integral method (BIM). This method utilizes the strong points of FDM (i.e. localized stencil) and BIM (i.e. accurate representation of particle surface). Specifically, in each iteration, the flow field away from the particles is solved on a Cartesian FDM grid, while the traction on the particle surface (given the the velocity of the particle) is solved using BIM. The two schemes are coupled by matching the solution in an intermediate region between the particle and surrounding fluid. We validate this method by solving for flow around an array of cylinders, and find good agreement with Hasimoto's (J. Fluid Mech. 1959) analytical results.
Finite Difference Approach for Estimating the Thermal Conductivity by 6-point Crank-Nicolson Scheme
SU Ya-xin; YANG Xiang-xiang
2005-01-01
Based on inverse heat conduction theory, a theoretical model using 6-point Crank-Nicolson finite difference scheme was used to calculate the thermal conductivity from temperature distribution, which can be measured experimentally. The method is a direct approach of second-order and the key advantage of the present method is that it is not required a priori knowledge of the functional form of the unknown thermal conductivity in the calculation and the thermal parameters are estimated only according to the known temperature distribution. Two cases were numerically calculated and the influence of experimental deviation on the precision of this method was discussed. The comparison of numerical and analytical results showed good agreement.
A simple parallel prefix algorithm for compact finite-difference schemes
Sun, Xian-He; Joslin, Ronald D.
1993-01-01
A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are difficult to solve efficiently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel prefix (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is highly efficient on parallel machines. It consists of a prefix communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study was conducted to provide a simple truncation formula. Experimental results were measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel prefix algorithm is a good algorithm for the compact scheme on high-performance computers.
Mickens, Ronald E.
1996-01-01
A large class of physical phenomena can be modeled by evolution and wave type Partial Differential Equations (PDE). Few of these equations have known explicit exact solutions. Finite-difference techniques are a popular method for constructing discrete representations of these equations for the purpose of numerical integration. However, the solutions to the difference equations often contain so called numerical instabilities; these are solutions to the difference equations that do not correspond to any solution of the PDE's. For explicit schemes, the elimination of this behavior requires functional relations to exist between the time and space steps-sizes. We show that such functional relations can be obtained for certain PDE's by use of a positivity condition. The PDE's studied are the Burgers, Fisher, and linearized Euler equations.
Landing-gear noise prediction using high-order finite difference schemes
Liu, Wen; Wook Kim, Jae; Zhang, Xin; Angland, David; Caruelle, Bastien
2013-07-01
Aerodynamic noise from a generic two-wheel landing-gear model is predicted by a CFD/FW-H hybrid approach. The unsteady flow-field is computed using a compressible Navier-Stokes solver based on high-order finite difference schemes and a fully structured grid. The calculated time history of the surface pressure data is used in an FW-H solver to predict the far-field noise levels. Both aerodynamic and aeroacoustic results are compared to wind tunnel measurements and are found to be in good agreement. The far-field noise was found to vary with the 6th power of the free-stream velocity. Individual contributions from three components, i.e. wheels, axle and strut of the landing-gear model are also investigated to identify the relative contribution to the total noise by each component. It is found that the wheels are the dominant noise source in general. Strong vortex shedding from the axle is the second major contributor to landing-gear noise. This work is part of Airbus LAnding Gear nOise database for CAA validatiON (LAGOON) program with the general purpose of evaluating current CFD/CAA and experimental techniques for airframe noise prediction.
Hammer, René; Arnold, Anton
2013-01-01
A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stabi...
Mickens, Ronald E.
1989-01-01
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
Dimitrov, Yuri M.; Vulkov, Lubin G.
2016-01-01
We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed.
Martowicz, A.; Ruzzene, M.; Staszewski, W. J.; Rimoli, J. J.; Uhl, T.
2014-03-01
The work deals with the reduction of numerical dispersion in simulations of wave propagation in solids. The phenomenon of numerical dispersion naturally results from time and spatial discretization present in a numerical model of mechanical continuum. Although discretization itself makes possible to model wave propagation in structures with complicated geometries and made of different materials, it inevitably causes simulation errors when improper time and length scales are chosen for the simulations domains. Therefore, by definition, any characteristic parameter for spatial and time resolution must create limitations on maximal wavenumber and frequency for a numerical model. It should be however noted that expected increase of the model quality and its functionality in terms of affordable wavenumbers, frequencies and speeds should not be achieved merely by denser mesh and reduced time integration step. The computational cost would be simply unacceptable. The authors present a nonlocal finite difference scheme with the coefficients calculated applying a Fourier series, which allows for considerable reduction of numerical dispersion. There are presented the results of analyses for 2D models, with isotropic and anisotropic materials, fulfilling the planar stress state. Reduced numerical dispersion is shown in the dispersion surfaces for longitudinal and shear waves propagating for different directions with respect to the mesh orientation and without dramatic increase of required number of nonlocal interactions. A case with the propagation of longitudinal wave in composite material is studied with given referential solution of the initial value problem for verification of the time-domain outcomes. The work gives a perspective of modeling of any type of real material dispersion according to measurements and with assumed accuracy.
Fisher, Travis C.; Carpenter, Mark H.; Nordstroem, Jan; Yamaleev, Nail K.; Swanson, R. Charles
2011-01-01
Simulations of nonlinear conservation laws that admit discontinuous solutions are typically restricted to discretizations of equations that are explicitly written in divergence form. This restriction is, however, unnecessary. Herein, linear combinations of divergence and product rule forms that have been discretized using diagonal-norm skew-symmetric summation-by-parts (SBP) operators, are shown to satisfy the sufficient conditions of the Lax-Wendroff theorem and thus are appropriate for simulations of discontinuous physical phenomena. Furthermore, special treatments are not required at the points that are near physical boundaries (i.e., discrete conservation is achieved throughout the entire computational domain, including the boundaries). Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and included in E. Narrow-stencil difference operators for linear viscous terms are also derived; these guarantee the conservative form of the combined operator.
Application of the 3D finite difference scheme to the TEXTOR-DED geometry
Zagorski, R.; Stepniewski, W. [Institute of Plasma Physics and Laser Microfusion, EURATOM Association, 01-497 Warsaw (Poland); Jakubowski, M. [Institut fuer Plasmaphysik, Forschungszentrum Juelich GmbH, EURATOM Association, Trilateral Euregio Cluster, D-52425 Juelich (Germany); McTaggart, N. [Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD (United Kingdom); Schneider, R.; Xanthopoulos, P. [Max-Planck-Institut fuer Plasmaphysik, Teilinstitut Greifswald, EURATOM Association, Wendelsteinstrasse 1, D-17491 Greifswald (Germany)
2006-09-15
In this paper, we use the finite difference code FINITE, developed for the stellarator geometry, to investigate the energy transport in the 3D TEXTOR-DED tokamak configuration. In particular, we concentrate on the comparison between two different algorithms for solving the radial part of the electron energy transport equation. (orig.)
Mashayekhi, Sima; Hugger, Jens
2015-01-01
market. In this paper, we compare several finite difference methods for the solution of this model with respect to precision and order of convergence within a computationally feasible domain allowing at most 200 space steps and 10000 time steps. We conclude that standard explicit Euler comes out as the...
Fuhrman, David R.; Bingham, Harry B.; Madsen, Per A.;
2004-01-01
rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water non-linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only...... insight into the numerical behaviour of this rather complicated system of non-linear PDEs.......This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...
Harte, Philip T.
1994-01-01
Proper discretization of a ground-water-flow field is necessary for the accurate simulation of ground-water flow by models. Although discretiza- tion guidelines are available to ensure numerical stability, current guidelines arc flexible enough (particularly in vertical discretization) to allow for some ambiguity of model results. Testing of two common types of vertical-discretization schemes (horizontal and nonhorizontal-model-layer approach) were done to simulate sloping hydrogeologic units characteristic of New England. Differences of results of model simulations using these two approaches are small. Numerical errors associated with use of nonhorizontal model layers are small (4 percent). even though this discretization technique does not adhere to the strict formulation of the finite-difference method. It was concluded that vertical discretization by means of the nonhorizontal layer approach has advantages in representing the hydrogeologic units tested and in simplicity of model-data input. In addition, vertical distortion of model cells by this approach may improve the representation of shallow flow processes.
A staggered mesh finite difference scheme for the computation of compressible flows
Sanders, Richard
1992-01-01
A simple high resolution finite difference technique is presented to approximate weak solutions to hyperbolic systems of conservation laws. The method does not rely on Riemann problem solvers and is therefore easy to extend to a wide variety of problems. The overall performance (resolution and CPU requirements) is competitive, with other state-of-the-art techniques offering sharp nonoscillatory shocks and contacts. Theoretical results confirm the reliability of the approach for linear systems and nonlinear scalar equations.
Direct Simulations of Transition and Turbulence Using High-Order Accurate Finite-Difference Schemes
Rai, Man Mohan
1997-01-01
In recent years the techniques of computational fluid dynamics (CFD) have been used to compute flows associated with geometrically complex configurations. However, success in terms of accuracy and reliability has been limited to cases where the effects of turbulence and transition could be modeled in a straightforward manner. Even in simple flows, the accurate computation of skin friction and heat transfer using existing turbulence models has proved to be a difficult task, one that has required extensive fine-tuning of the turbulence models used. In more complex flows (for example, in turbomachinery flows in which vortices and wakes impinge on airfoil surfaces causing periodic transitions from laminar to turbulent flow) the development of a model that accounts for all scales of turbulence and predicts the onset of transition may prove to be impractical. Fortunately, current trends in computing suggest that it may be possible to perform direct simulations of turbulence and transition at moderate Reynolds numbers in some complex cases in the near future. This seminar will focus on direct simulations of transition and turbulence using high-order accurate finite-difference methods. The advantage of the finite-difference approach over spectral methods is that complex geometries can be treated in a straightforward manner. Additionally, finite-difference techniques are the prevailing methods in existing application codes. In this seminar high-order-accurate finite-difference methods for the compressible and incompressible formulations of the unsteady Navier-Stokes equations and their applications to direct simulations of turbulence and transition will be presented.
Navnit Jha; R. K. Mohanty; Vinod Chauhan
2014-01-01
An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method h...
Nikkar, Samira; Nordström, Jan
2015-01-01
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discre...
On the impact of boundary conditions on dual consistent finite difference discretizations
Berg, Jens; Nordström, Jan
2013-01-01
In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier{Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed. The form of the boundary conditions is chosen such that reduction to rst order form with its complications can be avoided. The primal equation is discretized using ...
Computational Aero-Acoustic Using High-order Finite-Difference Schemes
Zhu, Wei Jun; Shen, Wen Zhong; Sørensen, Jens Nørkær
2007-01-01
In this paper, a high-order technique to accurately predict flow-generated noise is introduced. The technique consists of solving the viscous incompressible flow equations and inviscid acoustic equations using a incompressible/compressible splitting technique. The incompressible flow equations are solved using the in-house flow solver EllipSys2D/3D which is a second-order finite volume code. The acoustic solution is found by solving the acoustic equations using high-order finite difference sc...
To explore the behavior of electromagnetic waves in cold magnetized plasma, a three-dimensional cylindrical hybrid finite-difference time-domain model is developed. The full discrete dispersion relation is derived and compared with the exact solutions. We establish an analytical proof of stability in the case of nonmagnetized plasma. We demonstrate that in the case of nonmagnetized cold plasma the maximum stable Courant number of the hybrid method coincides with the vacuum Courant condition. In the case of magnetized plasma the stability of the applied numerical scheme is investigated by numerical simulation. In order to determine the utility of the applied difference scheme we complete the analysis of the numerical method demonstrating the limit of the reliability of the numerical results. (paper)
A convergent 2D finite-difference scheme for the Dirac-Poisson system and the simulation of graphene
Brinkman, Daniel
2014-01-01
We present a convergent finite-difference scheme of second order in both space and time for the 2D electromagnetic Dirac equation. We apply this method in the self-consistent Dirac-Poisson system to the simulation of graphene. The model is justified for low energies, where the particles have wave vectors sufficiently close to the Dirac points. In particular, we demonstrate that our method can be used to calculate solutions of the Dirac-Poisson system where potentials act as beam splitters or Veselago lenses. © 2013 Elsevier Inc.
Numerical pricing of options using high-order compact finite difference schemes
Tangman, D. Y.; Gopaul, A.; Bhuruth, M.
2008-09-01
We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black-Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.
Batista, F. B.; Albuquerque, E. L.; Arruda, J. R. F.; Dias, M.
2009-03-01
It is known that the elastic constants of composite materials can be identified by modal analysis and numerical methods. This approach is nondestructive, since it consists of simple tests and does not require high computational effort. It can be applied to isotropic, orthotropic, or anisotropic materials, making it a useful alternative for the characterization of composite materials. However, when elastic constants are bending constants, the method requires numerical spatial derivatives of experimental mode shapes. These derivatives are highly sensitive to noise. Previous works attempted to overcome the problem by using special optical devices. In this study, the elastic constant is identified using mode shapes obtained by standard laser vibrometers. To minimize errors, the mode shapes are first smoothed by regressive discrete Fourier series, after which their spatial derivatives are computed using finite differences. Numerical simulations using the finite element method and experimental results confirm the accuracy of the proposed method. The experimental examples reported here consist of an isotropic steel plate and an orthotropic carbon-epoxy plate excited with an electromechanical shaker. The forced response is measured at a large number of points, using a laser Doppler vibrometer. Both numerical and experimental results were satisfactory.
Navnit Jha
2014-01-01
Full Text Available An efficient algorithm for the numerical solution of higher (even orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy.
Wang, Cheng; Dong, XinZhuang; Shu, Chi-Wang
2015-10-01
For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.
Kiessling, Jonas
2014-05-06
Option prices in exponential Lévy models solve certain partial integro-differential equations. This work focuses on developing novel, computable error approximations for a finite difference scheme that is suitable for solving such PIDEs. The scheme was introduced in (Cont and Voltchkova, SIAM J. Numer. Anal. 43(4):1596-1626, 2005). The main results of this work are new estimates of the dominating error terms, namely the time and space discretisation errors. In addition, the leading order terms of the error estimates are determined in a form that is more amenable to computations. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschitz continuous as in previous works. If the underlying Lévy process has infinite jump activity, then the jumps smaller than some (Formula presented.) are approximated by diffusion. The resulting diffusion approximation error is also estimated, with leading order term in computable form, as well as the dependence of the time and space discretisation errors on this approximation. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the cut off parameter (Formula presented.). © 2014 Springer Science+Business Media Dordrecht.
Wei, Xiao-Kun; Shao, Wei; Shi, Sheng-Bing; Zhang, Yong; Wang, Bing-Zhong
2015-07-01
An efficient conformal locally one-dimensional finite-difference time-domain (LOD-CFDTD) method is presented for solving two-dimensional (2D) electromagnetic (EM) scattering problems. The formulation for the 2D transverse-electric (TE) case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit (ADI) CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field (TF/SF) boundary and the perfectly matched layer (PML), the radar cross section (RCS) of two 2D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method. Project supported by the National Natural Science Foundation of China (Grant Nos. 61331007 and 61471105).
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
Zhan, Ge
2013-02-19
The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. © 2013 Sinopec Geophysical Research Institute.
Two modified discrete chirp Fourier transform schemes
樊平毅; 夏香根
2001-01-01
This paper presents two modified discrete chirp Fourier transform (MDCFT) schemes.Some matched filter properties such as the optimal selection of the transform length, and its relationship to analog chirp-Fourier transform are studied. Compared to the DCFT proposed previously, theoretical and simulation results have shown that the two MDCFTs can further improve the chirp rate resolution of the detected signals.
The aim of the present work is to compare and discuss the three of the most advanced two dimensional transport methods, the finite difference and nodal discrete ordinates and surface flux method, incorporated into the transport codes TWODANT, TWOTRAN-NODAL, MULTIMEDIUM and SURCU. For intercomparison the eigenvalue and the neutron flux distribution are calculated using these codes in the LWR pool reactor benchmark problem. Additionally the results are compared with some results obtained by French collision probability transport codes MARSYAS and TRIDENT. Because the transport solution of this benchmark problem is close to its diffusion solution some results obtained by the finite element diffusion code FINELM and the finite difference diffusion code DIFF-2D are included
Campbell, W.
1981-01-01
A theoretical evaluation of the stability of an explicit finite difference solution of the transient temperature field in a composite medium is presented. The grid points of the field are assumed uniformly spaced, and media interfaces are either vertical or horizontal and pass through grid points. In addition, perfect contact between different media (infinite interfacial conductance) is assumed. A finite difference form of the conduction equation is not valid at media interfaces; therefore, heat balance forms are derived. These equations were subjected to stability analysis, and a computer graphics code was developed that permitted determination of a maximum time step for a given grid spacing.
El-Amin, M.F.
2011-05-14
In this paper, a finite difference scheme is developed to solve the unsteady problem of combined heat and mass transfer from an isothermal curved surface to a porous medium saturated by a non-Newtonian fluid. The curved surface is kept at constant temperature and the power-law model is used to model the non-Newtonian fluid. The explicit finite difference method is used to solve simultaneously the equations of momentum, energy and concentration. The consistency of the explicit scheme is examined and the stability conditions are determined for each equation. Boundary layer and Boussinesq approximations have been incorporated. Numerical calculations are carried out for the various parameters entering into the problem. Velocity, temperature and concentration profiles are shown graphically. It is found that as time approaches infinity, the values of wall shear, heat transfer coefficient and concentration gradient at the wall, which are entered in tables, approach the steady state values.
Steger, J. L.; Dougherty, F. C.; Benek, J. A.
1983-01-01
A mesh system composed of multiple overset body-conforming grids is described for adapting finite-difference procedures to complex aircraft configurations. In this so-called 'chimera mesh,' a major grid is generated about a main component of the configuration and overset minor grids are used to resolve all other features. Methods for connecting overset multiple grids and modifications of flow-simulation algorithms are discussed. Computational tests in two dimensions indicate that the use of multiple overset grids can simplify the task of grid generation without an adverse effect on flow-field algorithms and computer code complexity.
Settle, Sean O.
2013-01-01
The primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi- and nonuniform grids yields at most fourth- and third-order local accuracy, respectively. © 2013 Society for Industrial and Applied Mathematics.
Madavan, Nateri K.
1995-01-01
The work in this report was conducted at NASA Ames Research Center during the period from August 1993 to January 1995 deals with the direct numerical simulation of transitional and turbulent flow at low Mach numbers using high-order-accurate finite-difference techniques. A computation of transition to turbulence of the spatially-evolving boundary layer on a heated flat plate in the presence of relatively high freestream turbulence was performed. The geometry and flow conditions were chosen to match earlier experiments. The development of the momentum and thermal boundary layers was documented. Velocity and temperature profiles, as well as distributions of skin friction, surface heat transfer rate, Reynolds shear stress, and turbulent heat flux were shown to compare well with experiment. The numerical method used here can be applied to complex geometries in a straightforward manner.
We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes
On the accuracy and efficiency of finite difference solutions for nonlinear waves
Bingham, Harry B.
2006-01-01
We consider the relative accuracy and efficiency of low- and high-order finite difference discretizations of the exact potential flow problem for nonlinear water waves. The continuous differential operators are replaced by arbitrary order finite difference schemes on a structured but non...
Ackleh, Azmy S; Delcambre, Mark L; Sutton, Karyn L
2015-01-01
We present a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. This work extends the numerical theory and continues the preliminary studies on the model first developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721]. Numerical simulations demonstrating the accuracy of the method are presented, and we compare this scheme to the first-order scheme developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721] to show that the first-order method requires significantly more computational time to provide solutions with a similar accuracy. We also demonstrated that the model can be a tool to understand surprising or nonintuitive phenomena regarding competitive advantage in the context of biologically realistic growth, birth and death rates. PMID:25271885
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
Amirali, I.; Amiraliyev, G. M.; Cakir, M; Cimen, E.
2014-01-01
Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in...
An implicit finite-difference operator for the Helmholtz equation
Chu, Chunlei
2012-07-01
We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly. © 2012 Society of Exploration Geophysicists.
Delfin L, A.; Alonso V, G. [ININ, 52045 Ocoyoacac, Estado de Mexico (Mexico); Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: adl@nuclear.inin.mx
2003-07-01
In this work the development of a third order scheme of finite differences centered in mesh is presented and it is applied in the numerical solution of those diffusion equations in multi groups in stationary state and X Y geometry. Originally this scheme was developed by Hennart and del Valle for the monoenergetic diffusion equation with a well-known source and they show that the one scheme is of third order when comparing the numerical solution with the analytical solution of a model problem using several mesh refinements and boundary conditions. The scheme by them developed it also introduces the application of numeric quadratures to evaluate the rigidity matrices and of mass that its appear when making use of the finite elements method of Galerkin. One of the used quadratures is the open quadrature of 4 points, no-standard, of Newton-Cotes to evaluate in approximate form the elements of the rigidity matrices. The other quadrature is that of 3 points of Radau that it is used to evaluate the elements of all the mass matrices. One of the objectives of these quadratures are to eliminate the couplings among the Legendre moments 0 and 1 associated to the left and right faces as those associated to the inferior and superior faces of each cell of the discretization. The other objective is to satisfy the particles balance in weighed form in each cell. In this work it expands such development to multiplicative means considering several energy groups. There are described diverse details inherent to the technique, particularly those that refer to the simplification of the algebraic systems that appear due to the space discretization. Numerical results for several test problems are presented and are compared with those obtained with other nodal techniques. (Author)
Mimetic finite difference method
Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
2014-01-01
The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
Finite-Difference Algorithms For Computing Sound Waves
Davis, Sanford
1993-01-01
Governing equations considered as matrix system. Method variant of method described in "Scheme for Finite-Difference Computations of Waves" (ARC-12970). Present method begins with matrix-vector formulation of fundamental equations, involving first-order partial derivatives of primitive variables with respect to space and time. Particular matrix formulation places time and spatial coordinates on equal footing, so governing equations considered as matrix system and treated as unit. Spatial and temporal discretizations not treated separately as in other finite-difference methods, instead treated together by linking spatial-grid interval and time step via common scale factor related to speed of sound.
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
I. Amirali
2014-01-01
Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.
Explicit finite difference methods for the delay pseudoparabolic equations.
Amirali, I; Amiraliyev, G M; Cakir, M; Cimen, E
2014-01-01
Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown. PMID:24688392
Threshold Signature Scheme Based on Discrete Logarithm and Quadratic Residue
FEI Ru-chun; WANG Li-na
2004-01-01
Digital signature scheme is a very important research field in computer security and modern cryptography.A(k,n) threshold digital signature scheme is proposed by integrating digital signature scheme with Shamir secret sharing scheme.It can realize group-oriented digital signature, and its security is based on the difficulty in computing discrete logarithm and quadratic residue on some special conditions.In this scheme, effective digital signature can not be generated by any k-1 or fewer legal users, or only by signature executive.In addition, this scheme can identify any legal user who presents incorrect partial digital signature to disrupt correct signature, or any illegal user who forges digital signature.A method of extending this scheme to an Abelian group such as elliptical curve group is also discussed.The extended scheme can provide rapider computing speed and stronger security in the case of using shorter key.
Optimized Discretization Schemes For Brain Images
USHA RANI.N,
2011-02-01
Full Text Available In medical image processing active contour method is the important technique in segmenting human organs. Geometric deformable curves known as levelsets are widely used in segmenting medical images. In this modeling , evolution of the curve is described by the basic lagrange pde expressed as a function of space and time. This pde can be solved either using continuous functions or discrete numerical methods.This paper deals with the application of numerical methods like finite diffefence and TVd-RK methods for brain scans. The stability and accuracy of these methods are also discussed. This paper also deals with the more accurate higher order non-linear interpolation techniques like ENO and WENO in reconstructing the brain scans like CT,MRI,PET and SPECT is considered.
Design Validations for Discrete Logarithm Based Signature Schemes
Brickell, Ernest; Pointcheval, David; Vaudenay, Serge
2000-01-01
A number of signature schemes and standards have been recently designed, based on the discrete logarithm problem. Examples of standards are the DSA and the KCDSA. Very few formal design/security validations have already been conducted for both the KCDSA and the DSA, but in the "full" so-called random oracle model. In this paper we try to minimize the use of ideal hash functions for several Discrete Logarithm (DSS-like) signatures (abstracted as generic schemes). Namely, we show that the follo...
Optimal 25-Point Finite-Difference Subgridding Techniques for the 2D Helmholtz Equation
Tingting Wu
2016-01-01
Full Text Available We present an optimal 25-point finite-difference subgridding scheme for solving the 2D Helmholtz equation with perfectly matched layer (PML. This scheme is second order in accuracy and pointwise consistent with the equation. Subgrids are used to discretize the computational domain, including the interior domain and the PML. For the transitional node in the interior domain, the finite difference equation is formulated with ghost nodes, and its weight parameters are chosen by a refined choice strategy based on minimizing the numerical dispersion. Numerical experiments are given to illustrate that the newly proposed schemes can produce highly accurate seismic modeling results with enhanced efficiency.
An Efficient Signature Scheme based on Factoring and Discrete Logarithm
Ciss, Abdoul Aziz; Cheikh, Ahmed Youssef Ould
2012-01-01
This paper proposes a new signature scheme based on two hard problems : the cube root extraction modulo a composite moduli (which is equivalent to the factorisation of the moduli, IFP) and the discrete logarithm problem(DLP). By combining these two cryptographic assumptions, we introduce an efficient and strongly secure signature scheme. We show that if an adversary can break the new scheme with an algorithm $\\mathcal{A},$ then $\\mathcal{A}$ can be used to sove both the DLP and the IFP. The k...
Droniou, Jerome; Eymard,, Robert; Gallouët, Thierry; Herbin, Raphaele
2008-01-01
International audience We investigate the connections between several recent methods for the discretization of ani\\-so\\-tropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may ...
Beilina, Larisa
2016-08-01
We present domain decomposition finite element/finite difference method for the solution of hyperbolic equation. The domain decomposition is performed such that finite elements and finite differences are used in different subdomains of the computational domain: finite difference method is used on the structured part of the computational domain and finite elements on the unstructured part of the domain. Explicit discretizations for both methods are constructed such that the finite element and the finite difference schemes coincide on the common structured overlapping layer between computational subdomains. Then the resulting approach can be considered as a pure finite element scheme which avoids instabilities at the interfaces. We derive an energy estimate for the underlying hyperbolic equation with absorbing boundary conditions and illustrate efficiency of the domain decomposition method on the reconstruction of the conductivity function in three dimensions.
A Comparison of Continuous Mass-lumped Finite Elements and Finite Differences for 3D
Zhebel, E.; Minisini, S.; Kononov, A.; Mulder, W.A.
2012-01-01
The finite-difference method is widely used for time-domain modelling of the wave equation because of its ease of implementation of high-order spatial discretization schemes, parallelization and computational efficiency. However, finite elements on tetrahedral meshes are more accurate in complex geo
Accurate Finite Difference Algorithms
Goodrich, John W.
1996-01-01
Two families of finite difference algorithms for computational aeroacoustics are presented and compared. All of the algorithms are single step explicit methods, they have the same order of accuracy in both space and time, with examples up to eleventh order, and they have multidimensional extensions. One of the algorithm families has spectral like high resolution. Propagation with high order and high resolution algorithms can produce accurate results after O(10(exp 6)) periods of propagation with eight grid points per wavelength.
On Cryptographic Schemes Based on Discrete Logarithms and Factoring
Joye, Marc
At CRYPTO 2003, Rubin and Silverberg introduced the concept of torus-based cryptography over a finite field. We extend their setting to the ring of integers modulo N. We so obtain compact representations for cryptographic systems that base their security on the discrete logarithm problem and the factoring problem. This results in smaller key sizes and substantial savings in memory and bandwidth. But unlike the case of finite fields, analogous trace-based compression methods cannot be adapted to accommodate our extended setting when the underlying systems require more than a mere exponentiation. As an application, we present an improved, torus-based implementation of the ACJT group signature scheme.
Fa-yong Zhang
2004-01-01
The three-dimensional nonlinear Schrodinger equation with weakly damped that possesses a global attractor are considered. The dynamical properties of the discrete dynamical system which generate by a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete dynamical system.
Adaptive finite difference for seismic wavefield modelling in acoustic media.
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-01-01
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang's optimised finite difference scheme. PMID:27491333
Discrete unified gas kinetic scheme on unstructured meshes
Zhu, Lianhua; Xu, Kun
2015-01-01
The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for deterministic solution of the Boltzmann model equation with asymptotic preserving property. In DUGKS, the numerical flux of the distribution function is determined from a local numerical solution of the Boltzmann model equation using an unsplitting approach. The time step and mesh resolution are not restricted by the molecular collision time and mean free path. To demonstrate the capacity of DUGKS in practical problems, this paper extends the DUGKS to arbitrary unstructured meshes. Several tests of both internal and external flows are performed, which include the cavity flow ranging from continuum to free molecular regimes, a multiscale flow between two connected cavities with a pressure ratio of 10000, and a high speed flow over a cylinder in slip and transitional regimes. The numerical results demonstrate the effectiveness of the DUGKS in simulating multiscale flow problems.
There dimensional hydrodynamical calculations with heat transfer for nuclear reactors are complicated and actual tasks, their singularity is high numbers of Reynolds Re ∼ 106. The offered paper is one of initial development stages programs for problem solving the similar class. Operation contains exposition: mathematical setting of the task for the equations of Navier-Stokes with heat transfer compiling of space difference schemes by a method of check sizes, deriving of difference equations for pressure. The steady explicit methods of a solution of rigid tasks included in DUMKA program, and research of areas of their stability are used. Outcomes of numerical experiments of current of liquid in channels of rectangular cut are reduced. The complete spectrum analysis of the considered task is done (Authors)
Applications of an exponential finite difference technique
Handschuh, Robert F.; Keith, Theo G., Jr.
1988-01-01
An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.
Chin, Pius W. M.
2013-01-01
The optimal rate of convergence of the wave equation in both the energy and the ${L}^{2}$ -norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error in the ${L}^{2}$ -norm possesses the optimal rate of convergence $O\\left({h}^{...
Compatible discrete operator schemes on polyhedral meshes for elliptic and Stokes equations
This thesis presents a new class of spatial discretization schemes on polyhedral meshes, called Compatible Discrete Operator (CDO) schemes and their application to elliptic and Stokes equations In CDO schemes, preserving the structural properties of the continuous equations is the leading principle to design the discrete operators. De Rham maps define the degrees of freedom according to the physical nature of fields to discretize. CDO schemes operate a clear separation between topological relations (balance equations) and constitutive relations (closure laws). Topological relations are related to discrete differential operators, and constitutive relations to discrete Hodge operators. A feature of CDO schemes is the explicit use of a second mesh, called dual mesh, to build the discrete Hodge operator. Two families of CDO schemes are considered: vertex-based schemes where the potential is located at (primal) mesh vertices, and cell-based schemes where the potential is located at dual mesh vertices (dual vertices being in one-to-one correspondence with primal cells). The CDO schemes related to these two families are presented and their convergence is analyzed. A first analysis hinges on an algebraic definition of the discrete Hodge operator and allows one to identify three key properties: symmetry, stability, and P0-consistency. A second analysis hinges on a definition of the discrete Hodge operator using reconstruction operators, and the requirements on these reconstruction operators are identified. In addition, CDO schemes provide a unified vision on a broad class of schemes proposed in the literature (finite element, finite element, mimetic schemes... ). Finally, the reliability and the efficiency of CDO schemes are assessed on various test cases and several polyhedral meshes. (author)
FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION
Tie Zhang; Shuhua Zhang; Danmei Zhu
2009-01-01
In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the (O)(△t+△x2)-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.Mathematics subject classification: 65M12, 65M06, 91B28.
Accurate finite difference methods for time-harmonic wave propagation
Harari, Isaac; Turkel, Eli
1994-01-01
Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.
Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation
Kouatchou, Jules
1999-01-01
In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.
Simulation of Metasurfaces in Finite Difference Techniques
Vahabzadeh, Yousef; Caloz, Christophe
2016-01-01
We introduce a rigorous and simple method for analyzing metasurfaces, modeled as zero-thickness electromagnetic sheets, in Finite Difference (FD) techniques. The method consists in describing the spatial discontinuity induced by the metasurface as a virtual structure, located between nodal rows of the Yee grid, using a finite difference version of Generalized Sheet Transition Conditions (GSTCs). In contrast to previously reported approaches, the proposed method can handle sheets exhibiting both electric and magnetic discontinuities, and represents therefore a fundamental contribution in computational electromagnetics. It is presented here in the framework of the FD Frequency Domain (FDFD) method but also applies to the FD Time Domain (FDTD) scheme. The theory is supported by five illustrative examples.
Asymptotic analysis of discrete schemes for non-equilibrium radiation diffusion
Cui, Xia; Yuan, Guang-wei; Shen, Zhi-jun
2016-05-01
Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in [2] to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion.
A GOST-like Blind Signature Scheme Based on Elliptic Curve Discrete Logarithm Problem
HOSSEINI, Hossein; Bahrak, Behnam; Hessar, Farzad
2013-01-01
In this paper, we propose a blind signature scheme and three practical educed schemes based on elliptic curve discrete logarithm problem. The proposed schemes impart the GOST signature structure and utilize the inherent advantage of elliptic curve cryptosystems in terms of smaller key size and lower computational overhead to its counterpart public key cryptosystems such as RSA and ElGamal. The proposed schemes are proved to be secure and have less time complexity in comparison with the existi...
Exponential Finite-Difference Technique
Handschuh, Robert F.
1989-01-01
Report discusses use of explicit exponential finite-difference technique to solve various diffusion-type partial differential equations. Study extends technique to transient-heat-transfer problems in one dimensional cylindrical coordinates and two and three dimensional Cartesian coordinates and to some nonlinear problems in one or two Cartesian coordinates.
On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws
Bryson, Steve; Levy, Doron
2004-01-01
We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge-Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.
Static Analysis of Laminated Composite Plates by Finite Difference Method
Mustafa Haluk SARAÇOĞLU; Yunus ÖZÇELİKÖRS
2011-01-01
In this study; deflection at the mid-point of laminated composite rectangular plate subjected to uniformly distributed load is investigated by finite difference method. Four edges of these plates are Navier SS-1 simplysupported. Classical theory of laminated composite plates formed by extending the classical plate theory is used. Differential equations about bending of plate were discreted by finite difference method and unknown displacements at the related nodes are calculated. As an example...
The computer algebra approach of the finite difference methods for PDEs
In this paper, a first attempt has been made to realize the computer algebra construction of the finite difference methods or the finite difference schemes for constant coefficient partial differential equations. (author). 9 refs, 2 tabs
DESIGN OF A DIGITAL SIGNATURE SCHEME BASED ON FACTORING AND DISCRETE LOGARITHMS
杨利英; 覃征; 胡广伍; 王志敏
2004-01-01
Objective Focusing on the security problem of authentication and confidentiality in the context of computer networks, a digital signature scheme was proposed based on the public key cryptosystem. Methods Firstly, the course of digital signature based on the public key cryptosystem was given. Then, RSA and ELGamal schemes were described respectively. They were the basis of the proposed scheme. Generalized ELGamal type signature schemes were listed. After comparing with each other, one scheme, whose Signature equation was (m+r)x=j+s modΦ(p) , was adopted in the designing. Results Based on two well-known cryptographic assumptions, the factorization and the discrete logarithms, a digital signature scheme was presented. It must be required that s' was not equal to p'q' in the signing procedure, because attackers could forge the signatures with high probabilities if the discrete logarithms modulo a large prime were solvable. The variable public key "e" is used instead of the invariable parameter "3" in Harns signature scheme to enhance the security. One generalized ELGamal type scheme made the proposed scheme escape one multiplicative inverse operation in the signing procedure and one modular exponentiation in the verification procedure. Conclusion The presented scheme obtains the security that Harn's scheme was originally claimed. It is secure if the factorization and the discrete logarithms are simultaneously unsolvable.
Implicit finite-difference simulations of seismic wave propagation
Chu, Chunlei
2012-03-01
We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples. © 2012 Society of Exploration Geophysicists.
Zhou, Nanrun; Yang, Jianping; Tan, Changfa; Pan, Shumin; Zhou, Zhihong
2015-11-01
A new discrete fractional random transform based on two circular matrices is designed and a novel double-image encryption-compression scheme is proposed by combining compressive sensing with discrete fractional random transform. The two random circular matrices and the measurement matrix utilized in compressive sensing are constructed by using a two-dimensional sine Logistic modulation map. Two original images can be compressed, encrypted with compressive sensing and connected into one image. The resulting image is re-encrypted by Arnold transform and the discrete fractional random transform. Simulation results and security analysis demonstrate the validity and security of the scheme.
A New Digital Signature Scheme Based on Factoring and Discrete Logarithms
E. S. Ismail
2008-01-01
Full Text Available Problem statement: A digital signature scheme allows one to sign an electronic message and later the produced signature can be validated by the owner of the message or by any verifier. Most of the existing digital signature schemes were developed based on a single hard problem like factoring, discrete logarithm, residuosity or elliptic curve discrete logarithm problems. Although these schemes appear secure, one day in a near future they may be exploded if one finds a solution of the single hard problem. Approach: To overcome this problem, in this study, we proposed a new signature scheme based on multiple hard problems namely factoring and discrete logarithms. We combined the two problems into both signing and verifying equations such that the former depends on two secret keys whereas the latter depends on two corresponding public keys. Results: The new scheme was shown to be secure against the most five considering attacks for signature schemes. The efficiency performance of our scheme only requires 1203Tmul+Th time complexity for signature generation and 1202Tmul+Th time complexity for verification generation and this magnitude of complexity is considered minimal for multiple hard problems-like signature schemes. Conclusions: The new signature scheme based on multiple hard problems provides longer and higher security level than that scheme based on one problem. This is because no enemy can solve multiple hard problems simultaneously.
Discrete level schemes sublibrary. Progress report by Budapest group
An entirely new discrete levels file has been created by the Budapest group according to the recommended principles, using the Evaluated Nuclear Structure Data File, ENSDF as a source. The resulting library contains 96,834 levels and 105,423 gamma rays for 2,585 nuclei, with their characteristics such as energy, spin, parity, half-life as well gamma-ray energy and branching percentage
Symmetry-preserving discrete schemes for some heat transfer equations
Bakirova, Margarita; Dorodnitsyn, Vladimir; Kozlov, Roman
2004-01-01
Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterpa...
A. Caserta
1998-01-01
This paper deals with the antiplane wave propagation in a 2D heterogeneous dissipative medium with complex layer interfaces and irregular topography. The initial boundary value problem which represents the viscoelastic dynamics driving 2D antiplane wave propagation is formulated. The discretization scheme is based on the finite-difference technique. Our approach presents some innovative features. First, the introduction of the forcing term into the equation of motion offers the advantage of a...
Discrete unified gas kinetic scheme with force term for incompressible fluid flows
Wu, Chen; Chai, Zhenhua; Wang, Peng
2014-01-01
The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) method and unified gas kinetic scheme (UGKS) method, such as the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, DUGKS is proposed for near incompressible fluid flows, the existing compressible effect may cause some serious errors in simulating incompressible problems. To diminish the compressible effect, in this paper a novel DUGKS model with external force is developed for incompressible fluid flows by modifying the approximation of Maxwellian distribution. Meanwhile, due to the pressure boundary scheme, which is wildly used in many applications, has not been constructed for DUGKS, the non-equilibrium extrapolation (NEQ) scheme for both velocity and pressure boundary conditions is introduced. To illustrate the potential of the proposed model, numerical simul...
Compressed Semi-Discrete Central-Upwind Schemes for Hamilton-Jacobi Equations
Bryson, Steve; Kurganov, Alexander; Levy, Doron; Petrova, Guergana
2003-01-01
We introduce a new family of Godunov-type semi-discrete central schemes for multidimensional Hamilton-Jacobi equations. These schemes are a less dissipative generalization of the central-upwind schemes that have been recently proposed in series of works. We provide the details of the new family of methods in one, two, and three space dimensions, and then verify their expected low-dissipative property in a variety of examples.
TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS
Yidu Yang
2009-01-01
This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.
A mimetic finite difference method for the Stokes problem with elected edge bubbles
Lipnikov, K [Los Alamos National Laboratory; Berirao, L [DIPARTMENTO DI MATERMATICA
2009-01-01
A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The unstable P{sub 1}-P{sub 0} discretization is stabilized by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. The discretizations schemes for Stokes and Navier-Stokes equations must satisfy the celebrated inf-sup (or the LBB) stability condition. The stability condition implies a balance between discrete spaces for velocity and pressure. In finite elements, this balance is frequently achieved by adding bubble functions to the velocity space. The goal of this article is to show that the stabilizing edge bubble functions can be added only to a small set of mesh edges. This results in a smaller algebraic system and potentially in a faster calculations. We employ the mimetic finite difference (MFD) discretization technique that works for general polyhedral meshes and can accomodate non-uniform distribution of stabilizing bubbles.
M.-C. Casabán
2012-01-01
Full Text Available A new discretization strategy is introduced for the numerical solution of partial integrodifferential equations appearing in option pricing jump diffusion models. In order to consider the unknown behaviour of the solution in the unbounded part of the spatial domain, a double discretization is proposed. Stability, consistency, and positivity of the resulting explicit scheme are analyzed. Advantages of the method are illustrated with several examples.
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a
Thompson, K.G.
2000-11-01
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness
Generalized Rayleigh quotient and finite element two-grid discretization schemes
2009-01-01
This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.
Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case.
Guo, Zhaoli; Xu, Kun; Wang, Ruijie
2013-09-01
Based on the Boltzmann-BGK (Bhatnagar-Gross-Krook) equation, in this paper a discrete unified gas kinetic scheme (DUGKS) is developed for low-speed isothermal flows. The DUGKS is a finite-volume scheme with the discretization of particle velocity space. After the introduction of two auxiliary distribution functions with the inclusion of collision effect, the DUGKS becomes a fully explicit scheme for the update of distribution function. Furthermore, the scheme is an asymptotic preserving method, where the time step is only determined by the Courant-Friedricks-Lewy condition in the continuum limit. Numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current DUGKS. The comparison between the DUGKS and the well-defined lattice Boltzmann equation method (D2Q9) is presented as well. PMID:24125383
Using the Finite Difference Calculus to Sum Powers of Integers.
Zia, Lee
1991-01-01
Summing powers of integers is presented as an example of finite differences and antidifferences in discrete mathematics. The interrelation between these concepts and their analogues in differential calculus, the derivative and integral, is illustrated and can form the groundwork for students' understanding of differential and integral calculus.…
Computer-Oriented Calculus Courses Using Finite Differences.
Gordon, Sheldon P.
The so-called discrete approach in calculus instruction involves introducing topics from the calculus of finite differences and finite sums, both for motivation and as useful tools for applications of the calculus. In particular, it provides an ideal setting in which to incorporate computers into calculus courses. This approach has been…
Arbitrary Dimension Convection-Diffusion Schemes for Space-Time Discretizations
Bank, Randolph E. [Univ. of California, San Diego, CA (United States); Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Zikatanov, Ludmil T. [Bulgarian Academy of Sciences, Sofia (Bulgaria)
2016-01-20
This note proposes embedding a time dependent PDE into a convection-diffusion type PDE (in one space dimension higher) with singularity, for which two discretization schemes, the classical streamline-diffusion and the EAFE (edge average finite element) one, are investigated in terms of stability and error analysis. The EAFE scheme, in particular, is extended to be arbitrary order which is of interest on its own. Numerical results, in combined space-time domain demonstrate the feasibility of the proposed approach.
On some fundamental finite difference inequalities
B. G. Pachpatte
2001-01-01
The main object of this paper is to establish some new finite difference inequalities which can be used as tools in the study of various problems in the theory of certain classes of finite difference and sum-difference equations.
Spatial Parallelism of a 3D Finite Difference, Velocity-Stress Elastic Wave Propagation Code
MINKOFF,SUSAN E.
1999-12-09
Finite difference methods for solving the wave equation more accurately capture the physics of waves propagating through the earth than asymptotic solution methods. Unfortunately. finite difference simulations for 3D elastic wave propagation are expensive. We model waves in a 3D isotropic elastic earth. The wave equation solution consists of three velocity components and six stresses. The partial derivatives are discretized using 2nd-order in time and 4th-order in space staggered finite difference operators. Staggered schemes allow one to obtain additional accuracy (via centered finite differences) without requiring additional storage. The serial code is most unique in its ability to model a number of different types of seismic sources. The parallel implementation uses the MP1 library, thus allowing for portability between platforms. Spatial parallelism provides a highly efficient strategy for parallelizing finite difference simulations. In this implementation, one can decompose the global problem domain into one-, two-, and three-dimensional processor decompositions with 3D decompositions generally producing the best parallel speed up. Because i/o is handled largely outside of the time-step loop (the most expensive part of the simulation) we have opted for straight-forward broadcast and reduce operations to handle i/o. The majority of the communication in the code consists of passing subdomain face information to neighboring processors for use as ''ghost cells''. When this communication is balanced against computation by allocating subdomains of reasonable size, we observe excellent scaled speed up. Allocating subdomains of size 25 x 25 x 25 on each node, we achieve efficiencies of 94% on 128 processors. Numerical examples for both a layered earth model and a homogeneous medium with a high-velocity blocky inclusion illustrate the accuracy of the parallel code.
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
Walter H. Aschbacher
2009-01-01
Full Text Available We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications.
FULL DISCRETE TWO-LEVEL CORRECTION SCHEME FOR NAVIER-STOKES EQUATIONS
Yanren Hou; Liquan Mei
2008-01-01
In this paper,a full discrete two-level scheme for the unsteady Navier-Stokes equations based on a time dependent projection approach is proposed. In the sense of the new projection and its related space splitting,non-linearity is treated only on the coarse level subspace at each time step by solving exactly the standard Galerkin equation while a linear equation has to be solved on the fine level subspace to get the final approximation at this time step.Thus,it is a two-level based correction scheme for the standard Galerkin approximation.Stability and error estimate for this scheme are investigated in the paper.
Phonon Boltzmann equation-based discrete unified gas kinetic scheme for multiscale heat transfer
Guo, Zhaoli
2016-01-01
Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant-Friedrichs-Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.
Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs
Cuicui Liao
2012-01-01
discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.
HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS
Jianxian Qiu
2007-01-01
In this paper, we use Hermite weighted essentially non-oscillatory (HWENO) schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result,comparing with HWENO with Runge-Kutta time discretizations schemes (HWENO-RK) of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction.Extensive numerical experiments are performed to illustrate the capability of the method.
Static Analysis of Laminated Composite Plates by Finite Difference Method
Mustafa Haluk SARAÇOĞLU
2011-01-01
Full Text Available In this study; deflection at the mid-point of laminated composite rectangular plate subjected to uniformly distributed load is investigated by finite difference method. Four edges of these plates are Navier SS-1 simplysupported. Classical theory of laminated composite plates formed by extending the classical plate theory is used. Differential equations about bending of plate were discreted by finite difference method and unknown displacements at the related nodes are calculated. As an example; mid point dimensionless deflections of specially orhotropic, regular symmetric and regular antisymmetric composite laminated square plates under uniformly distributed load were examined.
Practical aspects of prestack depth migration with finite differences
Ober, C.C.; Oldfield, R.A.; Womble, D.E.; Romero, L.A. [Sandia National Labs., Albuquerque, NM (United States); Burch, C.C. [Conoco Inc. (United States)
1997-07-01
Finite-difference, prestack, depth migrations offers significant improvements over Kirchhoff methods in imaging near or under salt structures. The authors have implemented a finite-difference prestack depth migration algorithm for use on massively parallel computers which is discussed. The image quality of the finite-difference scheme has been investigated and suggested improvements are discussed. In this presentation, the authors discuss an implicit finite difference migration code, called Salvo, that has been developed through an ACTI (Advanced Computational Technology Initiative) joint project. This code is designed to be efficient on a variety of massively parallel computers. It takes advantage of both frequency and spatial parallelism as well as the use of nodes dedicated to data input/output (I/O). Besides giving an overview of the finite-difference algorithm and some of the parallelism techniques used, migration results using both Kirchhoff and finite-difference migration will be presented and compared. The authors start out with a very simple Cartoon model where one can intuitively see the multiple travel paths and some of the potential problems that will be encountered with Kirchhoff migration. More complex synthetic models as well as results from actual seismic data from the Gulf of Mexico will be shown.
Staggered-Grid Finite Difference Method with Variable-Order Accuracy for Porous Media
Jinghuai Gao; Yijie Zhang
2013-01-01
The numerical modeling of wave field in porous media generally requires more computation time than that of acoustic or elastic media. Usually used finite difference methods adopt finite difference operators with fixed-order accuracy to calculate space derivatives for a heterogeneous medium. A finite difference scheme with variable-order accuracy for acoustic wave equation has been proposed to reduce the computation time. In this paper, we develop this scheme for wave equations in porous media...
Direct Finite-Difference Simulations Of Turbulent Flow
Rai, Man Mohan; Moin, Parviz
1991-01-01
Report discusses use of upwind-biased finite-difference numerical-integration scheme to simulate evolution of small disturbances and fully developed turbulence in three-dimensional flow of viscous, incompressible fluid in channel. Involves use of computational grid sufficiently fine to resolve motion of fluid at all relevant length scales.
A Review of High-Order and Optimized Finite-Difference Methods for Simulating Linear Wave Phenomena
Zingg, David W.
1996-01-01
This paper presents a review of high-order and optimized finite-difference methods for numerically simulating the propagation and scattering of linear waves, such as electromagnetic, acoustic, or elastic waves. The spatial operators reviewed include compact schemes, non-compact schemes, schemes on staggered grids, and schemes which are optimized to produce specific characteristics. The time-marching methods discussed include Runge-Kutta methods, Adams-Bashforth methods, and the leapfrog method. In addition, the following fourth-order fully-discrete finite-difference methods are considered: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. For each method studied, the number of grid points per wavelength required for accurate simulation of wave propagation over large distances is presented. Recommendations are made with respect to the suitability of the methods for specific problems and practical aspects of their use, such as appropriate Courant numbers and grid densities. Avenues for future research are suggested.
An energy conserving finite-difference model of Maxwell's equations for soliton propagation
Bachiri, H; Vázquez, L
1997-01-01
We present an energy conserving leap-frog finite-difference scheme for the nonlinear Maxwell's equations investigated by Hile and Kath [C.V.Hile and W.L.Kath, J.Opt.Soc.Am.B13, 1135 (96)]. The model describes one-dimensional scalar optical soliton propagation in polarization preserving nonlinear dispersive media. The existence of a discrete analog of the underlying continuous energy conservation law plays a central role in the global accuracy of the scheme and a proof of its generalized nonlinear stability using energy methods is given. Numerical simulations of initial fundamental, second and third-order hyperbolic secant soliton pulses of fixed spatial full width at half peak intensity containing as few as 4 and 8 optical carrier wavelengths, confirm the stability, accuracy and efficiency of the algorithm. The effect of a retarded nonlinear response time of the media modeling Raman scattering is under current investigation in this context.
Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation
Koley, U
2011-01-01
We are concerned with the convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier- Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.
Peng Wang
2013-01-01
Full Text Available When the conservative governing equation of incompressible fluid flow and heat transfer is discretized by the finite volume method, there are various schemes to deal with the convective term. In this paper, studies on the convective term discretized by two different schemes, named strong and weak conservation schemes, respectively, are presented in detail. With weak conservation scheme, the convective flux at interface is obtained by respective interpolation and subsequent product of primitive variables. With strong conservation scheme, the convective flux is treated as single physical variable for interpolation. The numerical results of two convection heat transfer cases indicate that under the same computation conditions, discretizing the convective term by strong conservation scheme would not only obtain a more accurate solution, but also guarantee the stability of computation and the clear physical meaning of the solution. Especially in the computation regions with sharp gradients, the advantages of strong conservation scheme become more apparent.
LIU Zhengxian; WANG Xuejun; DAI Jishuang; ZHANG Chuhua
2009-01-01
A second-order mixing difference scheme with a limiting factor is deduced with the reconstruction gra-dient method and applied to discretizing the Navier-Stokes equation in an unstructured grid. The transform of non-orthogonal diffusion items generated by the scheme in discrete equations is provided. The Delaunay triangulation method is improved to generate the unstructured grid. The computing program based on the SIMPLE algorithm in an unstructured grid is compiled and used to solve the discrete equations of two types of incompressible viscous flow. The numerical simulation results of the laminar flow driven by lid in cavity and flow behind a cylinder are compared with the theoretical solution and experimental data respectively. In the former case, a good agreement is achieved in the main velocity and drag coefficient curve. In the latter case, the numerical structure and development of vortex under several Reynolds numbers match well with that of the experiment. It is indicated that the factor dif-ference scheme is of higher accuracy, and feasible to be applied to Navier-Stokes equation.
The mimetic finite difference method for elliptic problems
Veiga, Lourenço Beirão; Manzini, Gianmarco
2014-01-01
This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
C.C. Stolk
2016-01-01
We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a discrete operator to be applied to the source and the wavefields
A Price-Based Demand Response Scheme for Discrete Manufacturing in Smart Grids
Zhe Luo
2016-08-01
Full Text Available Demand response (DR is a key technique in smart grid (SG technologies for reducing energy costs and maintaining the stability of electrical grids. Since manufacturing is one of the major consumers of electrical energy, implementing DR in factory energy management systems (FEMSs provides an effective way to manage energy in manufacturing processes. Although previous studies have investigated DR applications in process manufacturing, they were not conducted for discrete manufacturing. In this study, the state-task network (STN model is implemented to represent a discrete manufacturing system. On this basis, a DR scheme with a specific DR algorithm is applied to a typical discrete manufacturing—automobile manufacturing—and operational scenarios are established for the stamping process of the automobile production line. The DR scheme determines the optimal operating points for the stamping process using mixed integer linear programming (MILP. The results show that parts of the electricity demand can be shifted from peak to off-peak periods, reducing a significant overall energy costs without degrading production processes.
Oberman, Adam M.; Zwiers, Ian
2014-01-01
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equati...
İnan B.; Bahadir A. R.
2015-01-01
In this paper, numerical solutions of the generalized Burgers-Huxley equation are obtained using a new technique of forming improved exponential finite difference method. The technique is called implicit exponential finite difference method for the solution of the equation. Firstly, the implicit exponential finite difference method is applied to the generalized Burgers-Huxley equation. Since the generalized Burgers-Huxley equation is nonlinear the scheme leads to a system of nonlinear equatio...
A parallel adaptive finite difference algorithm for petroleum reservoir simulation
Hoang, Hai Minh
2005-07-01
Adaptive finite differential for problems arising in simulation of flow in porous medium applications are considered. Such methods have been proven useful for overcoming limitations of computational resources and improving the resolution of the numerical solutions to a wide range of problems. By local refinement of the computational mesh where it is needed to improve the accuracy of solutions, yields better solution resolution representing more efficient use of computational resources than is possible with traditional fixed-grid approaches. In this thesis, we propose a parallel adaptive cell-centered finite difference (PAFD) method for black-oil reservoir simulation models. This is an extension of the adaptive mesh refinement (AMR) methodology first developed by Berger and Oliger (1984) for the hyperbolic problem. Our algorithm is fully adaptive in time and space through the use of subcycling, in which finer grids are advanced at smaller time steps than the coarser ones. When coarse and fine grids reach the same advanced time level, they are synchronized to ensure that the global solution is conservative and satisfy the divergence constraint across all levels of refinement. The material in this thesis is subdivided in to three overall parts. First we explain the methodology and intricacies of AFD scheme. Then we extend a finite differential cell-centered approximation discretization to a multilevel hierarchy of refined grids, and finally we are employing the algorithm on parallel computer. The results in this work show that the approach presented is robust, and stable, thus demonstrating the increased solution accuracy due to local refinement and reduced computing resource consumption. (Author)
Frank, Jason
2006-01-01
In this note we show that multisymplectic Runge-Kutta box schemes, of which the Gauss-Legendre methods are the most important, preserve a discrete conservation law of wave action. The result follows by loop integration over an ensemble of flow realizations, and the local energy-momentum conservation law for continuous variables in semi-discretizations
A finite difference method for free boundary problems
Fornberg, Bengt
2010-04-01
Fornberg and Meyer-Spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a Cartesian grid. We show here how this procedure can be combined with a minimax-based optimization procedure to rapidly solve a wide range of elliptic-type free boundary value problems. © 2009 Elsevier B.V. All rights reserved.
High Order Finite Difference Methods for Multiscale Complex Compressible Flows
Sjoegreen, Bjoern; Yee, H. C.
2002-01-01
The classical way of analyzing finite difference schemes for hyperbolic problems is to investigate as many as possible of the following points: (1) Linear stability for constant coefficients; (2) Linear stability for variable coefficients; (3) Non-linear stability; and (4) Stability at discontinuities. We will build a new numerical method, which satisfies all types of stability, by dealing with each of the points above step by step.
Optimization of Dengue Epidemics: a test case with different discretization schemes
Rodrigues, Helena Sofia; Torres, Delfim F M; 10.1063/1.3241345
2010-01-01
The incidence of Dengue epidemiologic disease has grown in recent decades. In this paper an application of optimal control in Dengue epidemics is presented. The mathematical model includes the dynamic of Dengue mosquito, the affected persons, the people's motivation to combat the mosquito and the inherent social cost of the disease, such as cost with ill individuals, educations and sanitary campaigns. The dynamic model presents a set of nonlinear ordinary differential equations. The problem was discretized through Euler and Runge Kutta schemes, and solved using nonlinear optimization packages. The computational results as well as the main conclusions are shown.
Verma, Prabal Singh
2015-01-01
The dimensionally split reconstruction method as described by Kurganov et al.\\cite{kurganov-2000} is revisited for better understanding and a simple fourth order scheme is introduced to solve 3D hyperbolic conservation laws following dimension by dimension approach. Fourth order central weighted essentially non-oscillatory (CWENO) reconstruction methods have already been proposed to study multidimensional problems \\cite{lpr4,cs12}. In this paper, it is demonstrated that a simple 1D fourth order CWENO reconstruction method by Levy et al.\\cite{lpr7} provides fourth order accuracy for 3D hyperbolic nonlinear problems when combined with the semi-discrete scheme by Kurganov et al.\\cite{kurganov-2000} and fourth order Runge-Kutta method for time integration.
Finite-difference computations of rotor loads
Caradonna, F. X.; Tung, C.
1985-01-01
The current and future potential of finite difference methods for solving real rotor problems which now rely largely on empiricism are demonstrated. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advanced-ratio flight. Comparisons are made with experimental pressure data.
An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs
Warming, Robert F.; Beam, Richard M.
1990-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L (sub 2) stability on a finite domain.
Weighted Average Finite Difference Methods for Fractional Reaction-Subdiffusion Equation
Nasser Hassen SWEILAM
2014-04-01
Full Text Available In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Numerical test examples, figures, and comparisons have been presented for clarity.doi:10.14456/WJST.2014.50
Second-order accurate nonoscillatory schemes for scalar conservation laws
Huynh, Hung T.
1989-01-01
Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.
High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations
Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)
2002-01-01
We present the first fifth order, semi-discrete central upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Tadmor-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spacial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.
A non-linear constrained optimization technique for the mimetic finite difference method
Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Svyatskiy, Daniil [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Bertolazzi, Enrico [Univ. of Trento (Italy); Frego, Marco [Univ. of Trento (Italy)
2014-09-30
This is a strategy for the construction of monotone schemes in the framework of the mimetic finite difference method for the approximation of diffusion problems on unstructured polygonal and polyhedral meshes.
2D time-domain finite-difference modeling for viscoelastic seismic wave propagation
Fan, Na; Zhao, Lian-Feng; Xie, Xiao-Bi; Ge, Zengxi; Yao, Zhen-Xing
2016-07-01
Real Earth media are not perfectly elastic. Instead, they attenuate propagating mechanical waves. This anelastic phenomenon in wave propagation can be modeled by a viscoelastic mechanical model consisting of several standard linear solids. Using this viscoelastic model, we approximate a constant Q over a frequency band of interest. We use a four-element viscoelastic model with a tradeoff between accuracy and computational costs to incorporate Q into 2D time-domain first-order velocity-stress wave equations. To improve the computational efficiency, we limit the Q in the model to a list of discrete values between 2 and 1000. The related stress and strain relaxation times that characterize the viscoelastic model are pre-calculated and stored in a database for use by the finite-difference calculation. A viscoelastic finite-difference scheme that is second-order in time and fourth-order in space is developed based on the MacCormack algorithm. The new method is validated by comparing the numerical result with analytical solutions that are calculated using the generalized reflection/transmission coefficient method. The synthetic seismograms exhibit greater than 95 per cent consistency in a two-layer viscoelastic model. The dispersion generated from the simulation is consistent with the Kolsky-Futterman dispersion relationship.
Discrete unified gas kinetic scheme for all Knudsen number flows: II. Compressible case
Guo, Zhaoli; Xu, Kun
2014-01-01
This paper is a continuation of our earlier work [Z.L. Guo {\\it et al.}, Phys. Rev. E {\\bf 88}, 033305 (2013)] where a multiscale numerical scheme based on kinetic model was developed for low speed isothermal flows with arbitrary Knudsen numbers. In this work, a discrete unified gas-kinetic scheme (DUGKS) for compressible flows with the consideration of heat transfer and shock discontinuity is developed based on the Shakhov model with an adjustable Prandtl number. The method is an explicit finite-volume scheme where the transport and collision processes are coupled in the evaluation of the fluxes at cell interfaces, so that the nice asymptotic preserving (AP) property is retained, such that the time step is limited only by the CFL number, the distribution function at cell interface recovers to the Chapman-Enskog one in the continuum limit while reduces to that of free-transport for free-molecular flow, and the time and spatial accuracy is of second-order accuracy in smooth region. These features make the DUGK...
Marian Malec; Lucjan Sapa
2007-01-01
This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in \\(\\mathbf{R}^{1+n}\\). A suitable finite difference scheme is constructed. It is proved that the scheme has a unique sol...
罗振东; 朱江; 谢正辉; 张桂芳
2003-01-01
The non-stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non-stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.
Solutions of the System of Differential Equations by Differential Transform/Finite Difference Method
SÜNGÜ, İnci ÇİLİNGİR; DEMIR, Huseyin
2012-01-01
In this study, Differential Transform/Finite Difference Method is considered as a new solution technique. Discretization of system of first and second order linear and nonlinear differential equations were investigated and approximate solutions were compared with the solutions of Adomian Decomposition Method. The results show that Differential Transform/Finite Difference method is one of the efficient approaches to solve system of differential equations. Consequently, it was shown that the hy...
SMALL-STENCIL PAD(E) SCHEMES TO SOLVE NONLINEAR EVOLUTION EQUATIONS
LIU Ru-xun; WU Ling-ling
2005-01-01
A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.
The Leray- Gårding method for finite difference schemes
Coulombel, Jean-François
2015-01-01
International audience In [Ler53] and [ Går56], Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, Går56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the der...
On the wavelet optimized finite difference method
Jameson, Leland
1994-01-01
When one considers the effect in the physical space, Daubechies-based wavelet methods are equivalent to finite difference methods with grid refinement in regions of the domain where small scale structure exists. Adding a wavelet basis function at a given scale and location where one has a correspondingly large wavelet coefficient is, essentially, equivalent to adding a grid point, or two, at the same location and at a grid density which corresponds to the wavelet scale. This paper introduces a wavelet optimized finite difference method which is equivalent to a wavelet method in its multiresolution approach but which does not suffer from difficulties with nonlinear terms and boundary conditions, since all calculations are done in the physical space. With this method one can obtain an arbitrarily good approximation to a conservative difference method for solving nonlinear conservation laws.
Accurate Finite Difference Methods for Option Pricing
Persson, Jonas
2006-01-01
Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance level...
Halman, Nir; Klabjan, Diego; Mostagir, Mohamed; Orlin, Jim; Simchi-Levi, David
2009-01-01
The single-item stochastic inventory control problem is to find an inventory replenishment policy in the presence of independent discrete stochastic demands under periodic review and finite time horizon. In this paper, we prove that this problem is intractable and design for it a fully polynomial-time approximation scheme.
Numerical study on three-dimensional steady incompressible laminar flows in a square duct of 90 .deg. bend is undertaken to evaluate the accuracy of four different discretization schemes from lower-order to higher-order by a new solution code(PowerCFD) using unstructured cell-centered method. Detailed comparisons between computed solutions and available experimental data are given mainly for the velocity distributions at several cross-sections in a 90 deg. bend square duct with developed entry flows. Detailed comparisons are also made with several previous works using lower-order or higher-order schemes. Interesting features of the flow for each scheme are presented in detail
Non-linear analysis of skew thin plate by finite difference method
This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed
Deepak Sharma
2014-09-01
Full Text Available Encryption along with compression is the process used to secure any multimedia content processing with minimum data storage and transmission. The transforms plays vital role for optimizing any encryption-compression systems. Earlier the original information in the existing security system based on the fractional Fourier transform (FRFT is protected by only a certain order of FRFT. In this article, a novel method for encryption-compression scheme based on multiple parameters of discrete fractional Fourier transform (DFRFT with random phase matrices is proposed. The multiple-parameter discrete fractional Fourier transform (MPDFRFT possesses all the desired properties of discrete fractional Fourier transform. The MPDFRFT converts to the DFRFT when all of its order parameters are the same. We exploit the properties of multiple-parameter DFRFT and propose a novel encryption-compression scheme using the double random phase in the MPDFRFT domain for encryption and compression data. The proposed scheme with MPDFRFT significantly enhances the data security along with image quality of decompressed image compared to DFRFT and FRFT and it shows consistent performance with different images. The numerical simulations demonstrate the validity and efficiency of this scheme based on Peak signal to noise ratio (PSNR, Compression ratio (CR and the robustness of the schemes against bruit force attack is examined.
Kovács, M; Lindgren, F
2012-01-01
We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.
Kim, S. [Purdue Univ., West Lafayette, IN (United States)
1994-12-31
Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.
A parallel finite-difference method for computational aerodynamics
A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed. 14 refs
Numerical simulation of solitons in the nerve axon using finite differences
Werpers, Jonatan
2014-01-01
A High-order accurate finite difference scheme is derived for a non-linear soliton model of nerve signal propagation in axons. Boundary conditions yielding well-posed problems are suggested and included in the scheme using a penalty technique. Stability is shown using the summation-by-parts framework for a frozen parameter version of the non-linear problem.
The discrete variational derivative method based on discrete differential forms
Yaguchi, Takaharu; Matsuo, Takayasu; Sugihara, Masaaki
2012-05-01
As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete variational derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn-Hilliard equation and the Klein-Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H1-stable under some assumptions. In particular, one for the nonlinear Klein-Gordon equation is obtained by combination of the energy conservation property and the discrete Poincaré inequality, which are the temporal and spacial structures that are preserved by the above methods.
The Complex-Step-Finite-Difference method
Abreu, Rafael; Stich, Daniel; Morales, Jose
2015-07-01
We introduce the Complex-Step-Finite-Difference method (CSFDM) as a generalization of the well-known Finite-Difference method (FDM) for solving the acoustic and elastic wave equations. We have found a direct relationship between modelling the second-order wave equation by the FDM and the first-order wave equation by the CSFDM in 1-D, 2-D and 3-D acoustic media. We present the numerical methodology in order to apply the introduced CSFDM and show an example for wave propagation in simple homogeneous and heterogeneous models. The CSFDM may be implemented as an extension into pre-existing numerical techniques in order to obtain fourth- or sixth-order accurate results with compact three time-level stencils. We compare advantages of imposing various types of initial motion conditions of the CSFDM and demonstrate its higher-order accuracy under the same computational cost and dispersion-dissipation properties. The introduced method can be naturally extended to solve different partial differential equations arising in other fields of science and engineering.
Contraction pre-conditioner in finite-difference electromagnetic modelling
Yavich, Nikolay; Zhdanov, Michael S.
2016-09-01
This paper introduces a novel approach to constructing an effective pre-conditioner for finite-difference (FD) electromagnetic modelling in geophysical applications. This approach is based on introducing an FD contraction operator, similar to one developed for integral equation formulation of Maxwell's equation. The properties of the FD contraction operator were established using an FD analogue of the energy equality for the anomalous electromagnetic field. A new pre-conditioner uses a discrete Green's function of a 1-D layered background conductivity. We also developed the formulae for an estimation of the condition number of the system of FD equations pre-conditioned with the introduced FD contraction operator. Based on this estimation, we have established that the condition number is bounded by the maximum conductivity contrast between the background conductivity and actual conductivity. When there are both resistive and conductive anomalies relative to the background, the new pre-conditioner is advantageous over using the 1-D discrete Green's function directly. In our numerical experiments with both resistive and conductive anomalies, for a land geoelectrical model with 1:10 contrast, the method accelerates convergence of an iterative method (BiCGStab) by factors of 2-2.5, and in a marine example with 1:50 contrast, by a factor of 4.6, compared to direct use of the discrete 1-D Green's function as a pre-conditioner.
Finite difference methods for coupled flow interaction transport models
Shelly McGee
2009-04-01
Full Text Available Understanding chemical transport in blood flow involves coupling the chemical transport process with flow equations describing the blood and plasma in the membrane wall. In this work, we consider a coupled two-dimensional model with transient Navier-Stokes equation to model the blood flow in the vessel and Darcy's flow to model the plasma flow through the vessel wall. The advection-diffusion equation is coupled with the velocities from the flows in the vessel and wall, respectively to model the transport of the chemical. The coupled chemical transport equations are discretized by the finite difference method and the resulting system is solved using the additive Schwarz method. Development of the model and related analytical and numerical results are presented in this work.
Moteki, Nobuhiro
2016-07-01
An accurate and efficient simulation of light scattering by an atmospheric black carbon (BC)-containing aerosol-a fractal-like cluster of hundreds of carbon monomers that is internally mixed with other aerosol compounds such as sulfates, organics, and water-remains challenging owing to the enormous diversities of such aerosols' size, shape, and mixing state. Although the discrete dipole approximation (DDA) is theoretically an exact numerical method that is applicable to arbitrary non-spherical inhomogeneous targets, in practice, it suffers from severe granularity-induced error and degradation of computational efficiency for such extremely complex targets. To solve this drawback, we propose herein a hybrid DDA method designed for arbitrary BC-containing aerosols: the monomer-dipole assumption is applied to a cluster of carbon monomers, whereas the efficient cubic-lattice discretization is applied to the remaining particle volume consisting of other materials. The hybrid DDA is free from the error induced by the surface granularity of carbon monomers that occurs in conventional cubic-lattice DDA. In the hybrid DDA, we successfully mitigate the artifact of neglecting the higher-order multipoles in the monomer-dipole assumption by incorporating the magnetic dipole in addition to the electric dipole into our DDA formulations. Our numerical experiments show that the hybrid DDA method is an efficient light-scattering solver for BC-containing aerosols in arbitrary mixing states. The hybrid DDA could be also useful for a cluster of metallic nanospheres associated with other dielectric materials.
Efficient architectures for two-dimensional discrete wavelet transform using lifting scheme.
Xiong, Chengyi; Tian, Jinwen; Liu, Jian
2007-03-01
Novel architectures for 1-D and 2-D discrete wavelet transform (DWT) by using lifting schemes are presented in this paper. An embedded decimation technique is exploited to optimize the architecture for 1-D DWT, which is designed to receive an input and generate an output with the low- and high-frequency components of original data being available alternately. Based on this 1-D DWT architecture, an efficient line-based architecture for 2-D DWT is further proposed by employing parallel and pipeline techniques, which is mainly composed of two horizontal filter modules and one vertical filter module, working in parallel and pipeline fashion with 100% hardware utilization. This 2-D architecture is called fast architecture (FA) that can perform J levels of decomposition for N * N image in approximately 2N2(1 - 4(-J))/3 internal clock cycles. Moreover, another efficient generic line-based 2-D architecture is proposed by exploiting the parallelism among four subband transforms in lifting-based 2-D DWT, which can perform J levels of decomposition for N * N image in approximately N2(1 - 4(-J))/3 internal clock cycles; hence, it is called high-speed architecture. The throughput rate of the latter is increased by two times when comparing with the former 2-D architecture, but only less additional hardware cost is added. Compared with the works reported in previous literature, the proposed architectures for 2-D DWT are efficient alternatives in tradeoff among hardware cost, throughput rate, output latency and control complexity, etc. PMID:17357722
We describe a number of alternative iterative schemes to sweep the spatial grid in order to numerically solve one-speed X,Y-geometry neutron transport problems in the discrete ordinates (SN) formulation. We perform numerical experiments with the one-node block inversion iterative schemes to solve the discretized equations on the linear nodal method and we illustrate the computational performance of each iterative scheme for typical steady-state model problems. (author)
The sub-library of discrete level schemes and gamma radiation branching ratios (DLS) is translated from the evaluated nuclear structure data file (ENSDF). The data are further checked and corrected. In consideration of the demands for different kinds of research fields most of the evaluated experimental levels and their gamma rays in the ENSDF are kept in DLS data file. the management-retrieval code can provide two retrieving ways. One is a retrieval for a single nucleus (SN), and the other is one for a neutron reaction (NR). The latter contains four kinds of retrieving types corresponding four types of different fast neutron calculation codes. The code can cut off and select the required level and gamma rays from whole discrete level scheme according to user's demands
WANG Hong-yu; TIAN Zuo-hua; SHI Song-jiao; WENG Zheng-xin
2008-01-01
This paper proposes a robust fault detection and isolation (FDI) scheme for discrete time-delay system with disturbance. The FDI scheme can not only detect but also isolate the faults. The lifting method is exploited to transform the discrete time-delay system into the non-time-delay form. A generalized structured residual set is designed based on the unknown input observer (UIO). For each residual generator, one of the system input signals together with the corresponding actuator fault and the disturbance signals are treated as an unknown input term. The residual signals can not only be robust against the disturbance, but also be of the capacity to isolate the actuator faults. The proposed method has been verified by a numerical example.
Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation
Prentice, J. S. C.
2012-01-01
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…
Abstract Level Parallelization of Finite Difference Methods
Edwin Vollebregt
1997-01-01
Full Text Available A formalism is proposed for describing finite difference calculations in an abstract way. The formalism consists of index sets and stencils, for characterizing the structure of sets of data items and interactions between data items (“neighbouring relations”. The formalism provides a means for lifting programming to a more abstract level. This simplifies the tasks of performance analysis and verification of correctness, and opens the way for automaticcode generation. The notation is particularly useful in parallelization, for the systematic construction of parallel programs in a process/channel programming paradigm (e.g., message passing. This is important because message passing, unfortunately, still is the only approach that leads to acceptable performance for many more unstructured or irregular problems on parallel computers that have non-uniform memory access times. It will be shown that the use of index sets and stencils greatly simplifies the determination of which data must be exchanged between different computing processes.
Chu, Chunlei
2009-01-01
We analyze the dispersion properties and stability conditions of the high‐order convolutional finite difference operators and compare them with the conventional finite difference schemes. We observe that the convolutional finite difference method has better dispersion properties and becomes more efficient than the conventional finite difference method with the increasing order of accuracy. This makes the high‐order convolutional operator a good choice for anisotropic elastic wave simulations on rotated staggered grids since its enhanced dispersion properties can help to suppress the numerical dispersion error that is inherent in the rotated staggered grid structure and its efficiency can help us tackle 3D problems cost‐effectively.
A novel approach is presented in this paper for improving anisotropic diffusion PDE models, based on the Perona–Malik equation. A solution is proposed from an engineering perspective to adaptively estimate the parameters of the regularizing function in this equation. The goal of such a new adaptive diffusion scheme is to better preserve edges when the anisotropic diffusion PDE models are applied to image enhancement tasks. The proposed adaptive parameter estimation in the anisotropic diffusion PDE model involves self-organizing maps and Bayesian inference to define edge probabilities accurately. The proposed modifications attempt to capture not only simple edges but also difficult textural edges and incorporate their probability in the anisotropic diffusion model. In the context of the application of PDE models to image processing such adaptive schemes are closely related to the discrete image representation problem and the investigation of more suitable discretization algorithms using constraints derived from image processing theory. The proposed adaptive anisotropic diffusion model illustrates these concepts when it is numerically approximated by various discretization schemes in a database of magnetic resonance images (MRI), where it is shown to be efficient in image filtering and restoration applications
Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling
Karjalainen Matti
2004-01-01
Full Text Available Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one-dimensional (1D band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D and three-dimensional (3D mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented.
Viscoelastic Finite Difference Modeling Using Graphics Processing Units
Fabien-Ouellet, G.; Gloaguen, E.; Giroux, B.
2014-12-01
Full waveform seismic modeling requires a huge amount of computing power that still challenges today's technology. This limits the applicability of powerful processing approaches in seismic exploration like full-waveform inversion. This paper explores the use of Graphics Processing Units (GPU) to compute a time based finite-difference solution to the viscoelastic wave equation. The aim is to investigate whether the adoption of the GPU technology is susceptible to reduce significantly the computing time of simulations. The code presented herein is based on the freely accessible software of Bohlen (2002) in 2D provided under a General Public License (GNU) licence. This implementation is based on a second order centred differences scheme to approximate time differences and staggered grid schemes with centred difference of order 2, 4, 6, 8, and 12 for spatial derivatives. The code is fully parallel and is written using the Message Passing Interface (MPI), and it thus supports simulations of vast seismic models on a cluster of CPUs. To port the code from Bohlen (2002) on GPUs, the OpenCl framework was chosen for its ability to work on both CPUs and GPUs and its adoption by most of GPU manufacturers. In our implementation, OpenCL works in conjunction with MPI, which allows computations on a cluster of GPU for large-scale model simulations. We tested our code for model sizes between 1002 and 60002 elements. Comparison shows a decrease in computation time of more than two orders of magnitude between the GPU implementation run on a AMD Radeon HD 7950 and the CPU implementation run on a 2.26 GHz Intel Xeon Quad-Core. The speed-up varies depending on the order of the finite difference approximation and generally increases for higher orders. Increasing speed-ups are also obtained for increasing model size, which can be explained by kernel overheads and delays introduced by memory transfers to and from the GPU through the PCI-E bus. Those tests indicate that the GPU memory size
Cagnetti, Filippo
2013-11-01
We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(h) convergence rate in terms of the L∞ norm and O(h) in terms of the L1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. © 2013 IMACS.
Saarelma, Jukka; Botts, Jonathan; Hamilton, Brian; Savioja, Lauri
2016-04-01
Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%. PMID:27106330
SIMULATION OF POLLUTANTS IN RIVER SYSTEMS USING FINITE DIFFERENCE METHOD
ZAHEER Iqbal; CUI Guang Bai
2002-01-01
This paper using finite difference scheme for the numerical solution of advection-dispersion equation develops a one-dimensional water quality model. The model algorithm has some modification over other steady state models including QUAL2E, which have been used steady state implementation of implicit backward-difference numerical scheme. The computer program in the developed model contains a special unsteady state implementation of four point implicit upwind numerical schemes using double sweep method. The superiority of this method in the modeling procedure results the simulation efficacy under simplified conditions of effluent discharge from point and non-point sources. The model is helpful for eye view assessment of degree of interaction between model variables for strategic planning purposes. The model has been applied for the water quality simulation of the Hanjiang River basin using flow computation model. Model simulation results have shown the pollutants prediction, dispersion and impact on the existing water quality.Model test shows the model validity comparing with other sophisticated models. Sensitivity analysis was performed to overview the most sensitive parameters followed by calibration and verification process.
Finite difference analysis of the transient temperature profile within GHARR-1 fuel element
Highlights: • Transient heat conduction for GHARR-1 fuel was developed and simulated by MATLAB. • The temperature profile after shutdown showed parabolic decay pattern. • The recorded temperature of about 411.6 K was below the melting point of the clad. • The fuel is stable and no radioactivity will be released into the coolant. - Abstract: Mathematical model of the transient heat distribution within Ghana Research Reactor-1 (GHARR-1) fuel element and related shutdown heat generation rates have been developed. The shutdown heats considered were residual fission and fission product decay heat. A finite difference scheme for the discretization by implicit method was used. Solution algorithms were developed and MATLAB program implemented to determine the temperature distributions within the fuel element after shutdown due to reactivity insertion accident. The simulations showed a steady state temperature of about 341.3 K which deviated from that reported in the GHARR-1 safety analysis report by 2% error margin. The average temperature obtained under transient condition was found to be approximately 444 K which was lower than the melting point of 913 K for the aluminium cladding. Thus, the GHARR-1 fuel element was stable and there would be no release of radioactivity in the coolant during accident conditions
ON FINITE DIFFERENCES ON A STRING PROBLEM
J. M. Mango
2014-01-01
Full Text Available This study presents an analysis of a one-Dimensional (1D time dependent wave equation from a vibrating guitar string. We consider the transverse displacement of a plucked guitar string and the subsequent vibration motion. Guitars are known for production of great sound in form of music. An ordinary string stretched between two points and then plucked does not produce quality sound like a guitar string. A guitar string produces loud and unique sound which can be organized by the player to produce music. Where is the origin of guitar sound? Can the contribution of each part of the guitar to quality sound be accounted for, by mathematically obtaining the numerical solution to wave equation describing the vibration of the guitar string? In the present sturdy, we have solved the wave equation for a vibrating string using the finite different method and analyzed the wave forms for different values of the string variables. The results show that the amplitude (pitch or quality of the guitar wave (sound vary greatly with tension in the string, length of the string, linear density of the string and also on the material of the sound board. The approximate solution is representative; if the step width; ∂x and ∂t are small, that is <0.5.
Determination of finite-difference weights using scaled binomial windows
Chu, Chunlei
2012-05-01
The finite-difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes. Conventional finite-difference weights can be calculated either from Taylor series expansions or by Lagrange interpolation polynomials. The finite-difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain. For this reason, we also can derive finite-difference operators by truncating the convolution series of the pseudospectral method. Various truncation windows can be employed for this purpose and they result in finite-difference operators with different dispersion properties. We found that there exists two families of scaled binomial windows that can be used to derive conventional finite-difference operators analytically. With a minor change, these scaled binomial windows can also be used to derive optimized finite-difference operators with enhanced dispersion properties. © 2012 Society of Exploration Geophysicists.
High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves
Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter
2012-01-01
with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations....
New developments for increased performance of the SBP-SAT finite difference technique
Nordström, Jan; Eliasson, Peter
2015-01-01
In this article, recent developments for increased performance of the high order and stable SBP-SAT finite difference technique is described. In particularwe discuss the use ofweak boundary conditions and dual consistent formulations.The use ofweak boundary conditions focus on increased convergence to steady state, and hence efficiency. Dual consistent schemes produces superconvergent functionals and increases accuracy.
High Order Finite Difference Methods, Multidimensional Linear Problems and Curvilinear Coordinates
Nordstrom, Jan; Carpenter, Mark H.
1999-01-01
Boundary and interface conditions are derived for high order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.
High-order finite difference solution for 3D nonlinear wave-structure interaction
Ducrozet, Guillaume; Bingham, Harry B.; Engsig-Karup, Allan Peter;
2010-01-01
This contribution presents our recent progress on developing an efficient fully-nonlinear potential flow model for simulating 3D wave-wave and wave-structure interaction over arbitrary depths (i.e. in coastal and offshore environment). The model is based on a high-order finite difference scheme...
Komenda, Jan; Masopust, Tomáš; van Schuppen, J. H.
Berlin: The International Federation of Automatic Control, 2010 - (Raisch, J.; Giua, A.; Lafortune, S.; Moor, T.), s. 436-441 ISBN 978-3-902661-79-1. [10th International Workshop on Discrete Event Systems. Berlin (DE), 29.08.2010-01.09.2010] Grant ostatní: EU Projekt(XE) EU. ICT .DISC 224498 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete-event systems * modular supervisory control * coordinator * conditional controllability Subject RIV: BA - General Mathematics http://www.ifac-papersonline.net/Detailed/42964.html
Finite difference computing with exponential decay models
Langtangen, Hans Petter
2016-01-01
This text provides a very simple, initial introduction to the complete scientific computing pipeline: models, discretization, algorithms, programming, verification, and visualization. The pedagogical strategy is to use one case study – an ordinary differential equation describing exponential decay processes – to illustrate fundamental concepts in mathematics and computer science. The book is easy to read and only requires a command of one-variable calculus and some very basic knowledge about computer programming. Contrary to similar texts on numerical methods and programming, this text has a much stronger focus on implementation and teaches testing and software engineering in particular. .
Operto, S.; VIRIEUX, J; Ribodetti, Alessandra; Anderson, J E
2009-01-01
A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen's anisotropic parameters delta and epsilon, and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of l...
Sergei Turovets
2014-01-01
Full Text Available The Electrical Impedance Tomography (EIT and electroencephalography (EEG forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG- type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique.
Turovets, Sergei; Volkov, Vasily; Zherdetsky, Aleksej; Prakonina, Alena; Malony, Allen D
2014-01-01
The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique. PMID:24527060
Yassin, Ali A.
2014-01-01
Now, the security of digital images is considered more and more essential and fingerprint plays the main role in the world of image. Furthermore, fingerprint recognition is a scheme of biometric verification that applies pattern recognition techniques depending on image of fingerprint individually. In the cloud environment, an adversary has the ability to intercept information and must be secured from eavesdroppers. Unluckily, encryption and decryption functions are slow and they are often hard. Fingerprint techniques required extra hardware and software; it is masqueraded by artificial gummy fingers (spoof attacks). Additionally, when a large number of users are being verified at the same time, the mechanism will become slow. In this paper, we employed each of the partial encryptions of user's fingerprint and discrete wavelet transform to obtain a new scheme of fingerprint verification. Moreover, our proposed scheme can overcome those problems; it does not require cost, reduces the computational supplies for huge volumes of fingerprint images, and resists well-known attacks. In addition, experimental results illustrate that our proposed scheme has a good performance of user's fingerprint verification. PMID:27355051
An assessment of semi-discrete central schemes for hyperbolic conservation laws
High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in Z pinch physics. In this work, we describe initial efforts to develop a generalized 'black-box' conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between 'black-box' central schemes and the piecewise parabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM
Finite difference time domain analysis of chirped dielectric gratings
Hochmuth, Diane H.; Johnson, Eric G.
1993-01-01
The finite difference time domain (FDTD) method for solving Maxwell's time-dependent curl equations is accurate, computationally efficient, and straight-forward to implement. Since both time and space derivatives are employed, the propagation of an electromagnetic wave can be treated as an initial-value problem. Second-order central-difference approximations are applied to the space and time derivatives of the electric and magnetic fields providing a discretization of the fields in a volume of space, for a period of time. The solution to this system of equations is stepped through time, thus, simulating the propagation of the incident wave. If the simulation is continued until a steady-state is reached, an appropriate far-field transformation can be applied to the time-domain scattered fields to obtain reflected and transmitted powers. From this information diffraction efficiencies can also be determined. In analyzing the chirped structure, a mesh is applied only to the area immediately around the grating. The size of the mesh is then proportional to the electric size of the grating. Doing this, however, imposes an artificial boundary around the area of interest. An absorbing boundary condition must be applied along the artificial boundary so that the outgoing waves are absorbed as if the boundary were absent. Many such boundary conditions have been developed that give near-perfect absorption. In this analysis, the Mur absorbing boundary conditions are employed. Several grating structures were analyzed using the FDTD method.
An Implementable Scheme for Universal Lossy Compression of Discrete Markov Sources
Jalali, Shirin; Montanari, Andrea; Weissman, Tsachy
2009-01-01
We present a new lossy compressor for discrete sources. For coding a source sequence $x^n$, the encoder starts by assigning a certain cost to each reconstruction sequence. It then finds the reconstruction that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor. The cost of a sequence is given by a linear combination of its empirical probabilities of some order $k+1$ and its distortion relative to the source sequence. The linear structure of the ...
Discrete level schemes and their gamma radiation branching ratios (CENPL-DLS): Pt.2
The DLS data files contains the data and information of nuclear discrete levels and gamma rays. At present, it has 79461 levels and 93177 gamma rays for 1908 nuclides. The DLS sub-library has been set up at the CNDC, and widely used for nuclear model calculation and other field. the DLS management retrieval code DLS is introduced and an example is given for 56Fe. (1 tab.)
Matthews, Charles
2013-01-01
Molecular dynamics (MD) computations aim to simulate materials at the atomic level by approximating molecular interactions classically, relying on the Born-Oppenheimer approximation and semi-empirical potential energy functions as an alternative to solving the difficult time-dependent Schrodinger equation. An approximate solution is obtained by discretization in time, with an appropriate algorithm used to advance the state of the system between successive timesteps. Modern MD s...
Approximate Lie Group Analysis of Finite-difference Equations
Latypov, Azat M.
1995-01-01
Approximate group analysis technique, that is, the technique combining the methodology of group analysis and theory of small perturbations, is applied to finite-difference equations approximating ordinary differential equations. Finite-difference equations are viewed as a system of algebraic equations with a small parameter, introduced through the definitions of finite-difference derivatives. It is shown that application of the approximate invariance criterion to this algebraic system results...
Elements of Polya-Schur theory in finite difference setting
Brändén, P.; Krasikov, I.; Shapiro, B.
2012-01-01
In this note we attempt to develop an analog of P\\'olya-Schur theory describing the class of univariate hyperbolicity preservers in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e. the minimal distance between the roots) is at least one. In particular, finite difference versions of the classical Hermite-Poulain theorem and generalized Laguerre inequalities are obtained.
An Image Hiding Scheme Using 3D Sawtooth Map and Discrete Wavelet Transform
Ruisong Ye; Wenping Yu
2012-01-01
An image encryption scheme based on the 3D sawtooth map is proposed in this paper. The 3D sawtooth map is utilized to generate chaotic orbits to permute the pixel positions and to generate pseudo-random gray value sequences to change the pixel gray values. The image encryption scheme is then applied to encrypt the secret image which will be imbedded in one host image. The encrypted secret image and the host image are transformed by the wavelet transform and then are merged in the frequency d...
Reitman, L.; Wolfshtein, M.; Adler, D.
1982-11-01
A finite difference method is developed for solving the non-viscous formulation of a three-dimensional compressible flow problem for turbomachinery impellers. The numerical results and the time efficiency of this method are compared to that provided by a finite element method for this problem. The finite difference method utilizes a numerical, curvilinear, and non-orthogonal coordinate transformation and the ADI scheme. The finite difference method is utilized to solve a test problem of a centrifugal compressor impeller. It is shown that the finite difference method produces results in good agreement with the experimentally determined flow fields and is as accurate as the finite element technique. However, the finite difference method only requires about half the time in order to obtain the solution for this problem as that required by the finite element method.
Discrete Filters for Large Eddy Simulation of Forced Compressible MHD Turbulence
Chernyshov, Alexander A.; Karelsky, Kirill. V.; Petrosyan, Arakel. S.
2013-01-01
In present study, we discuss results of applicability of discrete filters for large eddy simulation (LES) method of forced compressible magnetohydrodynamic (MHD) turbulent flows with the scale-similarity model. Influences and effects of discrete filter shapes on the scale-similarity model are examined in physical space using a finite-difference numerical schemes. We restrict ourselves to the Gaussian filter and the top-hat filter. Representations of this subgrid-scale model which correspond t...
A finite difference, multipoint flux numerical approach to flow in porous media: Numerical examples
Osman, Hossam
2012-06-17
It is clear that none of the current available numerical schemes which may be adopted to solve transport phenomena in porous media fulfill all the required robustness conditions. That is while the finite difference methods are the simplest of all, they face several difficulties in complex geometries and anisotropic media. On the other hand, while finite element methods are well suited to complex geometries and can deal with anisotropic media, they are more involved in coding and usually require more execution time. Therefore, in this work we try to combine some features of the finite element technique, namely its ability to work with anisotropic media with the finite difference approach. We reduce the multipoint flux, mixed finite element technique through some quadrature rules to an equivalent cell-centered finite difference approximation. We show examples on using this technique to single-phase flow in anisotropic porous media.
On the Stability of the Finite Difference based Lattice Boltzmann Method
El-Amin, M.F.
2013-06-01
This paper is devoted to determining the stability conditions for the finite difference based lattice Boltzmann method (FDLBM). In the current scheme, the 9-bit two-dimensional (D2Q9) model is used and the collision term of the Bhatnagar- Gross-Krook (BGK) is treated implicitly. The implicitness of the numerical scheme is removed by introducing a new distribution function different from that being used. Therefore, a new explicit finite-difference lattice Boltzmann method is obtained. Stability analysis of the resulted explicit scheme is done using Fourier expansion. Then, stability conditions in terms of time and spatial steps, relaxation time and explicitly-implicitly parameter are determined by calculating the eigenvalues of the given difference system. The determined conditions give the ranges of the parameters that have stable solutions.
Optimal implicit 2-D finite differences to model wave propagation in poroelastic media
Itzá, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.
2016-05-01
Numerical modeling of seismic waves in heterogeneous porous reservoir rocks is an important tool for the interpretation of seismic surveys in reservoir engineering. We apply globally optimal implicit staggered-grid finite-differences to model 2-D wave propagation in heterogeneous poroelastic media at a low-frequency range (waves (for a porous media saturated with fluid). The numerical dispersion and stability conditions are derived using von Neumann analysis, showing that over a wide range of porous materials the Courant condition governs the stability and this optimal implicit scheme improves the stability of explicit schemes. High order explicit finite-differences (FD) can be replaced by some lower order optimal implicit FD so computational cost will not be as expensive while maintaining the accuracy. Here we compute weights for the optimal implicit FD scheme to attain an accuracy of γ = 10-8. The implicit spatial differentiation involves solving tridiagonal linear systems of equations through Thomas' algorithm.
A spherical higher-order finite-difference time-domain algorithm with perfectly matched layer
A higher-order finite-difference time-domain (HO-FDTD) in the spherical coordinate is presented in this paper. The stability and dispersion properties of the proposed scheme are investigated and an air-filled spherical resonator is modeled in order to demonstrate the advantage of this scheme over the finite-difference time-domain (FDTD) and the multiresolution time-domain (MRTD) schemes with respect to memory requirements and CPU time. Moreover, the Berenger's perfectly matched layer (PML) is derived for the spherical HO-FDTD grids, and the numerical results validate the efficiency of the PML. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
von Sydow, Lina
2013-10-01
The discontinuous Galerkin method for time integration of the Black-Scholes partial differential equation for option pricing problems is studied and compared with more standard time-integrators. In space an adaptive finite difference discretization is employed. The results show that the dG method are in most cases at least comparable to standard time-integrators and in some cases superior to them. Together with adaptive spatial grids the suggested pricing method shows great qualities.
An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation
Rongpei Zhang; Jia Liu; Guozhong Zhao
2015-01-01
We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our m...
Comparison Study on the Performances of Finite Volume Method and Finite Difference Method
Bo Yu(Brookhaven National Lab); Dongjie Wang; Xinyu Zhang; Wang Li; Renwei Liu
2013-01-01
Vorticity-stream function method and MAC algorithm are adopted to systemically compare the finite volume method (FVM) and finite difference method (FDM) in this paper. Two typical problems—lid-driven flow and natural convection flow in a square cavity—are taken as examples to compare and analyze the calculation performances of FVM and FDM with variant mesh densities, discrete forms, and treatments of boundary condition. It is indicated that FVM is superior to FDM from the pe...
Tam, Christopher K. W.; Webb, Jay C.
1994-01-01
In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The
Supervisory control synthesis of discrete-event systems using a coordination scheme
Komenda, Jan; Masopust, Tomáš; van Schuppen, J. H.
2012-01-01
Roč. 48, č. 2 (2012), s. 247-254. ISSN 0005-1098 R&D Projects: GA ČR(CZ) GAP103/11/0517; GA ČR GPP202/11/P028 Grant ostatní: European Commission(XE) EU.ICT.DISC 224498 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete-event systems * supervisory control * distributed control * closed-loop systems * controllability Subject RIV: BA - General Mathematics Impact factor: 2.919, year: 2012 http://www.sciencedirect.com/science/article/pii/S0005109811005395
On the modeling of the compressive behaviour of metal foams: a comparison of discretization schemes
Koudelka_ml., Petr; Zlámal, Petr; Kytýř, Daniel; Doktor, Tomáš; Fíla, Tomáš; Jiroušek, Ondřej
Kippen: Civil-Comp Press, 2013 - (Topping, B.; Iványi, P.). (Civil-Comp Proceedings. 102). ISBN 978-1-905088-57-7. ISSN 1759-3433. [International Conference on Civil, Structural and Environmental Engineering Computing /14./. Cagliari (IT), 03.09.2013-06.09.2013] R&D Projects: GA ČR(CZ) GAP105/12/0824 Institutional support: RVO:68378297 Keywords : aluminium foam * micromechanical properties * discretization * compressive behaviour * closed-cell geometry * microCT Subject RIV: JI - Composite Materials