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1

We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution.

Chalupecký, Vladimír

2011-01-01

2

On second-order mimetic and conservative finite-difference discretization schemes

Scientific Electronic Library Online (English)

Full Text Available SciELO Mexico | Language: English Abstract in spanish Aunque la derivación del esquema se puede realizar usando la reciente metodología de discretización numérica conocida como Diferencias Finitas Miméticas, estaremos presentando la derivación de un esquema de discretización mimético en diferencias finitas de segundo orden en una forma mas intuitiva, m [...] ediante el uso de expansiones de Taylor. Considerando que los estudiantes se familiarizan con expansiones de Taylor en los primeros cursos de cálculo y métodos matemáticos para físicos, pensamos que la presente alternativa de presentar este nuevo esquema de discretización es más favorable de ser asimilada en cursos de computación numérica tanto de pregrado como de postgrado. La robusticidad del esquema será ilustrada encontrando la solución numérica de un problema unidimensional del tipo capa límite difícil de resolver en forma numérica y que se basa en la ecuación de difusión estacionaria. Más aun, dado que el esquema de discretización alcanza segundo orden de precisión en todo el dominio computacional (incluyendo las fronteras), como ejercicio comparativo el mismo puede ser rápidamente aplicado para resolver ejemplos comúnmente encontrados en textos sobre métodos numéricos aplicados y que se resuelven usando otras metodologías numéricas (incluyendo algunos esquemas de discretización en diferencias finitas) Abstract in english Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansion [...] s. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to be more easily handled in courses on numerical computations of both undergraduate and graduated programs. The robustness of the resulting discretized equations will be illustrated by finding the numerical solution of an essentially hard-to-solve, one-dimensional, boundary-layer-like problem, based on the steady diffusion equation. Moreover, given that the presented mimetic discretization scheme attains second-order accuracy in the entire computational domain (including the boundaries), as a comparative exercise the discretized equations can be readily applied in solving examples commonly found in texbooks on applied numerical methods and solved numerically via other discretization schemes (including some of the standard finite-diffence discretization schemes)

S, Rojas; J.M, Guevara-Jordan.

2008-12-01

3

On second-order mimetic and conservative finite-difference discretization schemes

Scientific Electronic Library Online (English)

Full Text Available SciELO Mexico | Language: English Abstract in spanish Aunque la derivación del esquema se puede realizar usando la reciente metodología de discretización numérica conocida como Diferencias Finitas Miméticas, estaremos presentando la derivación de un esquema de discretización mimético en diferencias finitas de segundo orden en una forma mas intuitiva, m [...] ediante el uso de expansiones de Taylor. Considerando que los estudiantes se familiarizan con expansiones de Taylor en los primeros cursos de cálculo y métodos matemáticos para físicos, pensamos que la presente alternativa de presentar este nuevo esquema de discretización es más favorable de ser asimilada en cursos de computación numérica tanto de pregrado como de postgrado. La robusticidad del esquema será ilustrada encontrando la solución numérica de un problema unidimensional del tipo capa límite difícil de resolver en forma numérica y que se basa en la ecuación de difusión estacionaria. Más aun, dado que el esquema de discretización alcanza segundo orden de precisión en todo el dominio computacional (incluyendo las fronteras), como ejercicio comparativo el mismo puede ser rápidamente aplicado para resolver ejemplos comúnmente encontrados en textos sobre métodos numéricos aplicados y que se resuelven usando otras metodologías numéricas (incluyendo algunos esquemas de discretización en diferencias finitas) Abstract in english Although the scheme could be derived on the grounds of a relatively new numerical discretization methodology known as Mimetic Finite-Difference Approach, the derivation of a second-order mimetic finite difference discretization scheme will be presented in a more intuitive way, using Taylor expansion [...] s. Since students become familiar with Taylor expansions in earlier calculus and mathematical methods for physicist courses, one finds this approach of presenting this new discretization scheme to be more easily handled in courses on numerical computations of both undergraduate and graduated programs. The robustness of the resulting discretized equations will be illustrated by finding the numerical solution of an essentially hard-to-solve, one-dimensional, boundary-layer-like problem, based on the steady diffusion equation. Moreover, given that the presented mimetic discretization scheme attains second-order accuracy in the entire computational domain (including the boundaries), as a comparative exercise the discretized equations can be readily applied in solving examples commonly found in texbooks on applied numerical methods and solved numerically via other discretization schemes (including some of the standard finite-diffence discretization schemes)

S, Rojas; J.M, Guevara-Jordan.

4

Nonstandard finite difference schemes

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.

Mickens, Ronald E.

1995-01-01

5

Digital Repository Infrastructure Vision for European Research (DRIVER)

We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution.

Chalupecky?, V.; Muntean, A.

2011-01-01

6

TVD finite difference schemes and artificial viscosity

The total variation diminishing (TVD) finite difference scheme can be interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term. If a particular flux limiter is chosen and the requirement for upwind weighting is removed, an artificial dissipation term which is based on the theory of TVD schemes is obtained which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Numerical experiments to examine the performance of this new method are discussed.

Davis, S. F.

1984-01-01

7

Applications of nonstandard finite difference schemes

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

Mickens, Ronald E

2000-01-01

8

An optimized finite-difference scheme for wave propagation problems

International Nuclear Information System (INIS)

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. 16 refs

9

Finite-difference schemes for anisotropic diffusion

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 1012 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.

van Es, Bram; Koren, Barry; de Blank, Hugo J.

2014-09-01

10

Finite difference discretization of semiconductor drift-diffusion equations for nanowire solar cells

We introduce a finite difference discretization of semiconductor drift-diffusion equations using cylindrical partial waves. It can be applied to describe the photo-generated current in radial pn-junction nanowire solar cells. We demonstrate that the cylindrically symmetric (l=0) partial wave accurately describes the electronic response of a square lattice of silicon nanowires at normal incidence. We investigate the accuracy of our discretization scheme by using different mesh resolution along the radial direction r and compare with 3D (x, y, z) discretization. We consider both straight nanowires and nanowires with radius modulation along the vertical axis. The charge carrier generation profile inside each nanowire is calculated using an independent finite-difference time-domain simulation.

Deinega, Alexei; John, Sajeev

2012-10-01

11

In this paper we present and compare two unconditionally energy stable finite-difference schemes for the phase field crystal equation. The first is a one-step scheme based on a convex splitting of a discrete energy by Wise et al. [S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press]. In this scheme, which is first order in time and second order in space, the discrete energy is non-increasing for any time step. The second scheme we consider is a new, fully second-order two-step algorithm. In the new scheme, the discrete energy is bounded by its initial value for any time step. In both methods, the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable, regardless of time step-size. We solve the nonlinear equations using an efficient nonlinear multigrid method. Numerical simulations are presented and confirm the stability, efficiency and accuracy of the schemes.

Hu, Z.; Wise, S. M.; Wang, C.; Lowengrub, J. S.

2009-08-01

12

A generalized finite-difference time-domain scheme for solving nonlinear Schrödinger equations

Recently, we have developed a generalized finite-difference time-domain (G-FDTD) method for solving the time dependent linear Schrödinger equation. The G-FDTD is explicit and permits an accurate solution with simple computation, and also relaxes the stability condition as compared with the original FDTD scheme. In this article, we extend the G-FDTD scheme to solve nonlinear Schrödinger equations. Using the discrete energy method, the G-FDTD scheme is shown to satisfy a discrete analogous form of the conservation law. The obtained scheme is tested by three examples of soliton propagation, including bright and dark solitons as well as a 2D case. Compared with other popular existing methods, numerical results show that the present scheme provides a more accurate solution.

Moxley, Frederick Ira; Chuss, David T.; Dai, Weizhong

2013-08-01

13

ADI finite difference schemes for option pricing in the Heston model with correlation

This paper deals with the numerical solution of the Heston partial differential equation that plays an important role in financial option pricing, Heston (1993, Rev. Finan. Stud. 6). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on a non-uniform grid, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig & Sneyd scheme, the Modified Craig & Sneyd scheme, and the Hundsdorfer & Verwer sch...

Hout, K J in 't

2008-01-01

14

Approximation of systems of partial differential equations by finite difference schemes

International Nuclear Information System (INIS)

The approximation of Friedrichs' symmetric systems by a finite difference scheme with second order accuracy with respect to the step of discretization is studied. Unconditional stability of such a scheme is proved by the method of energy increase. This implicit scheme is then solved by three iterative methods: the first one, of the gradient type, converges slowly, the second one, of the Gauss-Seidel type, converges only if the system has been regularized to the first order with respect to the step of discretization by an elliptic operator, the last one, of the under-relaxation type, converges rapidly to a second order accurate solution. Explicit schemes for the integration of linear hyperbolic systems of evolution are considered. Conditional stability is proved for different schemes: Crank Nicolson, Leap-frog, Explicit, Predictor-corrector. Results relative to the explicit scheme are generalized to a quasi-linear, monotone system. Finally, stability and convergence in the solution of a finite difference scheme approximating an elliptic-parabolic equation, and an iterative method of relaxation for solving this scheme are studied. (author)

15

The nonlinear modified equation approach to analyzing finite difference schemes

The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.

Klopfer, G. H.; Mcrae, D. S.

1981-01-01

16

Computational Aero-Acoustic Using High-order Finite-Difference Schemes

International Nuclear Information System (INIS)

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 schemes. The incompressible flow equations and the acoustic equations are solved at the same time levels where the pressure and the velocities obtained from the incompressible equations form the input to the acoustic equations. To achieve low dissipation and dispersion errors, either Dispersion-Relation-Preserving (DRP) schemes or optimized compact finite difference schemes are used for spatial discretizations of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization

17

Computational Aero-Acoustic Using High-order Finite-Difference Schemes

DEFF Research Database (Denmark)

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 schemes. The incompressible flow equations and the acoustic equations are solved at the same time levels where the pressure and the velocities obtained from the incompressible equations form the input to the acoustic equations. To achieve low dissipation and dispersion errors, either Dispersion-Relation-Preserving (DRP) schemes or optimized compact finite difference schemes are used for spatial discretizations of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equationsfor time discretization.

Zhu, Wei Jun Technical University of Denmark,

2007-01-01

18

In this paper, a new 27-point finite difference method is presented for solving the 3D Helmholtz equation with perfectly matched layer (PML), which is a second order scheme and pointwise consistent with the equation. An error analysis is made between the numerical wavenumber and the exact wavenumber, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. A full-coarsening multigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear system stemming from the Helmholtz equation with PML by the finite difference scheme. The shifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectral analysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method, with a multigrid method used to invert the preconditioner approximately. Full-coarsening multigrid is employed, and a new matrix-based prolongation operator is constructed accordingly. Numerical results are presented to demonstrate the efficiency of both the new 27-point finite difference scheme with refined parameters, and the preconditioned Bi-CGSTAB method with the 3D full-coarsening multigrid.

Chen, Zhongying; Cheng, Dongsheng; Wu, Tingting

2012-10-01

19

Finite-Difference Lattice Boltzmann Scheme for High-Speed Compressible Flow: Two-Dimensional Case

International Nuclear Information System (INIS)

Lattice Boltzmann (LB) modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specific heats and a wide range of Mach number, from 0 to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 (2007) 502], a modified Lax-Wendroff finite difference scheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versus accuracy. The proposed model is validated by recovering results of some well-known benchmark tests: shock tubes and shock reflections. The new model may be used to track shock waves and/or to study the non-equilibrium procedure in the transition between the regular and Mach reflections of shock waves, etc

20

High-Order Finite Difference GLM-MHD Schemes for Cell-Centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting...

Mignone, A; Bodo, G

2010-01-01

21

An energy-stable finite-difference scheme for the binary fluid-surfactant system

We present an unconditionally energy stable finite-difference scheme for the binary fluid-surfactant system. The proposed method is based on the convex splitting of the energy functional with two variables. Here are two distinct features: (i) the convex splitting energy method is applied to energy functional with two variables, and (ii) the stability issue is related to the decay of the corresponding energy. The full discrete scheme leads to a decoupled system including a linear sub-system and a nonlinear sub-system. Algebraic multigrid and Newton-multigrid methods are adopted to solve the linear and nonlinear systems, respectively. Numerical experiments are shown to verify the stability of such a scheme.

Gu, Shuting; Zhang, Hui; Zhang, Zhengru

2014-08-01

22

Directory of Open Access Journals (Sweden)

Full Text Available A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.

Agus Suryanto

2012-06-01

23

Digital Repository Infrastructure Vision for European Research (DRIVER)

In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully ...

Baskaran, Arvind; Zhou, Peng; Hu, Zhengzheng; Wang, Cheng; Wise, Steven M.; Lowengrub, John S.

2012-01-01

24

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

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.

Gerritsen, Margot; Olsson, Pelle

1996-01-01

25

High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains

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.

Fisher, Travis C.; Carpenter, Mark H.

2013-01-01

26

Finite-difference scheme for the numerical solution of the Schroedinger equation

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.

Mickens, Ronald E.; Ramadhani, Issa

1992-01-01

27

A comparison of some implicit finite difference schemes used in flow boiling analysis

International Nuclear Information System (INIS)

This study has been limited to low order schemes which are already used in some system codes. We have tested them on the simplest two-phase flow model, i.e. the three equations homogeneous model. The main objective of the paper is to prove that the need for further comparisons is doubtless. We review briefly the difficulties which must be overcome for the comparisons to be valid. First of all, the discretization of the three conservation equations yields to non linear dicrete systems. The types of linearization used in codes such as RAMA, K. FIX and CLYSTERE are not equivalent at all and this might have a non-negligible infuence on the results and on the computer-time used to solve a particular problem. Secondly, for implicit schemes, the method to solve the linearized system should be the same, for instance Gaussian elimination with pivoting. Thirdly, special treatments for dealing with discontinuities such as the Boiling Boundary, the breech, if any, must be the same. If has been proven that some schemes can handle discontinuities much better than other and that's also an important point of comparison. These restrictions have convinced ourselves that a good comparison should be made with a unique code, specially designed to be very flexible with respect to the Finite Difference Scheme used

28

We present a fully conservative, skew-symmetric finite difference scheme on transformed grids. The skew-symmetry preserves the kinetic energy by first principles, simultaneously avoiding a central instability mechanism and numerical damping. In contrast to other skew-symmetric schemes no special averaging procedures are needed. Instead, the scheme builds purely on point-wise operations and derivatives. Any explicit and central derivative can be used, permitting high order and great freedom to optimize the scheme otherwise. This also allows the simple adaption of existing finite difference schemes to improve their stability and damping properties.

Reiss, Julius

2013-01-01

29

Development and application of a third order scheme of finite differences centered in mesh

International Nuclear Information System (INIS)

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)

30

International Nuclear Information System (INIS)

In this paper, a second-order Total Variation Diminishing (TVD) finite difference scheme of upwind type is employed for the numerical approximation of the classical hydrodynamic model for semiconductors proposed by Bloetekjaer and Baccarani-Wordeman. In particular, the high-order hyperbolic fluxes are evaluated by a suitable extrapolation on adjacent cells of the first-order fluxes of Roe, while total variation diminishing is achieved by limiting the slopes of the discrete Riemann invariants using the so-called Flux Corrected Transport approach. Extensive numerical simulations are performed on a submicron n+-n-n+ ballistic diode. The numerical experiments show that the spurious oscillations arising in the electron current are not completely suppressed by the TVD scheme, and can lead to serious numerical instabilities when the solution of the hydrodynamic model is non-smooth and the computational mesh is coarse. The accuracy of the numerical method is investigated in terms of conservation of the steady electron current. The obtained results show that the second-order scheme does not behave much better than a corresponding first-order one due to a poor performance of the slope limiters caused by the presence of local extrema of the Riemann invariant associated with the hyperbolic system

31

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 ...

32

A finite difference scheme for a degenerated diffusion equation arising in microbial ecology

Directory of Open Access Journals (Sweden)

Full Text Available A finite difference scheme is presented for a density-dependent diffusion equation that arises in the mathematical modelling of bacterial biofilms. The peculiarity of the underlying model is that it shows degeneracy as the dependent variable vanishes, as well as a singularity as the dependent variable approaches its a priori known upper bound. The first property leads to a finite speed of interface propagation if the initial data have compact support, while the second one introduces counter-acting super diffusion. This squeezing property of this model leads to steep gradients at the interface. Moving interface problems of this kind are known to be problematic for classical numerical methods and introduce non-physical and non-mathematical solutions. The proposed method is developed to address this observation. The central idea is a non-local (in time representation of the diffusion operator. It can be shown that the proposed method is free of oscillations at the interface, that the discrete interface satisfies a discrete version of the continuous interface condition and that the effect of interface smearing is quantitatively small.

Hermann J. Eberl

2007-02-01

33

In this work we develop a class of high-order finite difference weighted essentially non-oscillatory (FD-WENO) schemes for solving the ideal magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work is to use efficient high-order WENO spatial discretizations with high-order strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes. Numerical results have shown that with such methods we are able to resolve solution structures that are only visible at much higher grid resolutions with lower-order schemes. The key challenge in applying such methods to ideal MHD is to control divergence errors in the magnetic field. We achieve this by augmenting the base scheme with a novel high-order constrained transport approach that updates the magnetic vector potential. The predicted magnetic field from the base scheme is replaced by a divergence-free magnetic field that is obtained from the curl of this magnetic potential. The non-conservative weakly hyperbolic system that the magnetic vector potential satisfies is solved using a version of FD-WENO developed for Hamilton-Jacobi equations. The resulting numerical method is endowed with several important properties: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as point values on the same mesh (i.e., there is no mesh staggering); (2) both the spatial and temporal orders of accuracy are fourth-order; (3) no spatial integration or multidimensional reconstructions are needed in any step; and (4) special limiters in the magnetic vector potential update are used to control unphysical oscillations in the magnetic field. Several 2D and 3D numerical examples are presented to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks.

Christlieb, Andrew J.; Rossmanith, James A.; Tang, Qi

2014-07-01

34

The metrics and Jacobian in the fluid motion governing equations under curvilinear coordinate system have a variety of equivalent differential forms, which may have different discretization errors with the same difference scheme. The discretization errors of metrics and Jacobian may cause serious computational instability and inaccuracy in numerical results, especially for high-order finite difference schemes. It has been demonstrated by many researchers that the Geometric Conservation Law (GCL) is very important for high-order Finite Difference Methods (FDMs), and a proper form of metrics and Jacobian, which can satisfy the GCL, can considerably reduce discretization errors and computational instability. In order to satisfy the GCL for FDM, we have previously developed a Conservative Metric Method (CMM) to calculate the metrics [1] and the difference scheme ?3 in the CMM is determined with the suggestion ?3=?2. In this paper, a Symmetrical Conservative Metric Method (SCMM) is newly proposed based on the discussions of the metrics and Jacobian in FDM from geometry viewpoint by following the concept of vectorized surface and cell volume in Finite Volume Methods (FVMs). Interestingly, the expressions of metrics and Jacobian obtained by using the SCMM with second-order central finite difference scheme are equivalent to the vectorized surfaces and cell volumes, respectively. The main advantage of SCMM is that it makes the calculations based on high-order WCNS schemes aroud complex geometry flows possible and somewhat easy. Numerical tests on linear and nonlinear problems indicate that the quality of numerical results may be largely enhanced by utilizing the SCMM, and the advantage of the SCMM over other forms of metrics and Jacobian may be more evident on highly nonuniform grids.

Deng, Xiaogang; Min, Yaobing; Mao, Meiliang; Liu, Huayong; Tu, Guohua; Zhang, Hanxin

2013-04-01

35

The dispersion and dissipation properties of a scheme are of great importance for the simulation of flow fields which involve a broad range of length scales. In order to improve the spectral properties of the finite difference scheme, the authors have previously proposed the idea of optimizing the dispersion and dissipation properties separately and a fourth order scheme based on the minimized dispersion and controllable dissipation (MDCD) technique is thus constructed [29]. In the present paper, we further investigate this technique and extend it to a sixth order finite difference scheme to solve the Euler and Navier-Stokes equations. The dispersion properties of the scheme is firstly optimized by minimizing an elaborately designed integrated error function. Then the dispersion-dissipation condition which is newly derived by Hu and Adams [30] is introduced to supply sufficient dissipation to damp the unresolved wavenumbers. Furthermore, the optimized scheme is blended with an optimized Weighted Essentially Non-Oscillation (WENO) scheme to make it possible for the discontinuity-capturing. In this process, the approximation-dispersion-relation (ADR) approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several benchmark test problems, which include broadband fluctuations and strong shock waves, are solved to validate the high-resolution, the good discontinuity-capturing capability and the high-efficiency of the proposed scheme.

Sun, Zhen-sheng; Luo, Lei; Ren, Yu-xin; Zhang, Shi-ying

2014-08-01

36

Digital Repository Infrastructure Vision for European Research (DRIVER)

We give some results concerning the real-interpolation method and finite differences. Next, we apply them to estimate the resolvents of finite-difference discretizations of Dirichlet boundary value problems for elliptic equations in space dimensions one and two in analogs of spaces of continuous and HÃƒÂ¶lder continuous functions. Such results were employed to study finite-difference discretizations of parabolic equations.

Sergei Piskarev

2003-01-01

37

Directory of Open Access Journals (Sweden)

Full Text Available We give some results concerning the real-interpolation method and finite differences. Next, we apply them to estimate the resolvents of finite-difference discretizations of Dirichlet boundary value problems for elliptic equations in space dimensions one and two in analogs of spaces of continuous and HÃƒÂ¶lder continuous functions. Such results were employed to study finite-difference discretizations of parabolic equations.

Sergei Piskarev

2003-12-01

38

A consistent scheme for computation of the Reynolds stress and turbulent kinetic energy budgets is proposed for incompressible turbulent flow simulation using the finite difference method on a nonuniform staggered grid system. The present scheme is derived from the Navier-Stokes equation discretized by using the second-order energy-conservative finite differential method. The Reynolds stress and turbulent kinetic energy budgets in a fully developed channel flow computed by using the proposed schemes are compared with those by using the consistent scheme proposed for a uniform grid [Suzuki and Kawamura, Trans. JSME/B, Vol. 60, No. 578 (1994), pp. 3280-3286] and an intuitive scheme. The residuals in the Reynolds normal stress and turbulent kinetic energy budgets computed by using the present scheme and Suzuki-Kawamura scheme are found to be both sufficiently small. We also apply the same idea to the discretization of Reynolds shear stress budget, although the consistency is imperfect, be derived the numerical test shows that the residual is slightly smaller than that computed by the other schemes.

Mamori, Hiroya; Fukagata, Koji

39

Digital Repository Infrastructure Vision for European Research (DRIVER)

Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi-step integration methods. The coefficients are reverse engineered based on samples from a target function and its derivative used for training. The Runge-Kutta schemes are trained using the order condition equations. An appealing feature of the evolutionary ...

Erdbrink, C. D.; Krzhizhanovskaya, V. V.; Sloot, P. M. A.

2013-01-01

40

A new finite difference scheme for a dissipative cubic nonlinear Schrödinger equation

International Nuclear Information System (INIS)

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)

41

Accuracy of spectral and finite difference schemes in 2D advection problems

DEFF Research Database (Denmark)

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 implement, and applicable to complex geometries, but are somewhat inferior in accuracy compared to spectral schemes. Using two study cases at high Reynolds number, the merging of two equally signed Gaussian vortices in a periodic box and dipole interaction with a no-slip wall, we will demonstrate that the accuracy of FD schemes can be significantly improved if one is careful in choosing an appropriate FD scheme that reflects conservation properties of the nonlinear terms and in setting up the grid in accordance with the problem.

Naulin, V.; Nielsen, A.H.

2003-01-01

42

Boosting the Accuracy of Finite Difference Schemes via Optimal Time Step Selection

In this article, we present a novel technique for boosting the order of accuracy of formally low-order finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution. This technique allows us to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection on a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.

Chu, Kevin T

2008-01-01

43

International Nuclear Information System (INIS)

A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors

44

Digital Repository Infrastructure Vision for European Research (DRIVER)

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi and Zhang model (KPZ) and the Lai, Das Sarma and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal...

Gallego, Rafael; Castro, Mario; Lo?pez, Juan M.

2007-01-01

45

An initial-boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.

Zlotnik, I. A.

2011-03-01

46

Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation

International Nuclear Information System (INIS)

An explicit fourth-order finite-difference time-domain (FDTD) scheme using the symplectic integrator is applied to electromagnetic simulation. A feasible numerical implementation of the symplectic FDTD (SFDTD) scheme is specified. In particular, new strategies for the air-dielectric interface treatment and the near-to-far-field (NFF) transformation are presented. By using the SFDTD scheme, both the radiation and the scattering of three-dimensional objects are computed. Furthermore, the energy-conserving characteristic hold for the SFDTD scheme is verified under long-term simulation. Numerical results suggest that the SFDTD scheme is more efficient than the traditional FDTD method and other high-order methods, and can save computational resources

47

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

Directory of Open Access Journals (Sweden)

Full Text Available 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 efficient, reliable and robust technique for the numerical solutions of Burgers’ equation.

Vineet K. Srivastava

2013-08-01

48

Numerical solutions of coupled Burgers' equations by an implicit finite-difference scheme

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 efficient, reliable and robust technique for the numerical solutions of Burgers' equation.

Srivastava, Vineet K.; Singh, Sarita; Awasthi, Mukesh K.

2013-08-01

49

2D numerical simulation of the MEP energy-transport model with a finite difference scheme

A finite difference scheme of Scharfetter-Gummel type is used to simulate a consistent energy-transport model for electron transport in semiconductors devices, free of any fitting parameters, formulated on the basis of the maximum entropy principle. Simulations of silicon n +-n-n + diodes, 2D-MESFET and 2D-MOSFET and comparisons with the results obtained by a direct simulation of the Boltzmann transport equation and with other energy-transport models, known in the literature, show the validity of the model and the robustness of the numerical scheme.

Romano, V.

2007-02-01

50

In finite-difference (FD) acoustic forward modelling, the parameter settings for the perfectly matched layer (PML) are case-dependent. There is no explicit PML formula that can be applied for most acoustic models without tuning, especially, for the fourth-order FD scheme. In this paper, we propose an explicit PML formula for the acoustic frequency-domain FD with second-order and fourth-order accuracies, respectively. The fourth-order FD scheme uses a special treatment for the boundary. The number of points in the PML is fixed to be 10 and 15 for the second-order and fourth-order FD schemes, respectively. The maximum artificial attenuation parameter associated with the PML formula is automatically calculated based on the FD grid size and the value of the compressional velocity of the boundary cells of the interior domain. Numerical tests confirm that this empirical formula achieves the desired accuracy for 2-D and 3-D media for grid sizes varying from 1 to 200 m. For the fourth-order FD scheme, the proposed PML formula works effectively up to 25 points per wavelength for both 2-D and 3-D media. Beyond that, the error of the PML discretization becomes larger than the discretization error in the interior domain. For such cases and to keep a fourth-order accuracy, a larger number of points in the PML (thicker PML region) needs to be employed.

Pan, Guangdong; Abubakar, Aria; Habashy, Tarek M.

2012-01-01

51

A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics

International Nuclear Information System (INIS)

The authors present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio-Wu Riemann problems, the two-dimensional Kelvin-Helmholtz instability problems, and the two-dimensional Orszag-Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points

52

Directory of Open Access Journals (Sweden)

Full Text Available Numerical methods are extremely useful in solving real-life problems with complex materials and geometries. However, numerical methods in the time domain suffer from artificial numerical dispersion. Standard numerical techniques which are second-order in space and time, like the conventional Finite Difference 3-point (FD3 method, Finite-Difference Time-Domain (FDTD method, and Finite Integration Technique (FIT provide estimates of the error of discretized numerical operators rather than the error of the numerical solutions computed using these operators. Here optimally accurate time-domain FD operators which are second-order in time as well as in space are derived. Optimal accuracy means the greatest attainable accuracy for a particular type of scheme, e.g., second-order FD, for some particular grid spacing. The modified operators lead to an implicit scheme. Using the first order Born approximation, this implicit scheme is transformed into a two step explicit scheme, namely predictor-corrector scheme. The stability condition (maximum time step for a given spatial grid interval for the various modified schemes is roughly equal to that for the corresponding conventional scheme. The modified FD scheme (FDM attains reduction of numerical dispersion almost by a factor of 40 in 1-D case, compared to the FD3, FDTD, and FIT. The CPU time for the FDM scheme is twice of that required by the FD3 method. The simulated synthetic data for a 2-D P-SV (elastodynamics problem computed using the modified scheme are 30 times more accurate than synthetics computed using a conventional scheme, at a cost of only 3.5 times as much CPU time. The FDM is of particular interest in the modeling of large scale (spatial dimension is more or equal to one thousand wave lengths or observation time interval is very high compared to reference time step wave propagation and scattering problems, for instance, in ultrasonic antenna and synthetic scattering data modeling for Non-Destructive Testing (NDT applications, where other standard numerical methods fail due to numerical dispersion effects. The possibility of extending this method to staggered grid approach is also discussed. The numerical FD3, FDTD, FIT, and FDM results are compared against analytical solutions.

C. Bommaraju

2005-01-01

53

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid ...

Lombard, Bruno; Gélis, Céline; Virieux, Jean

2007-01-01

54

In order to solve high-resolution 3D problems in computational geodynamics it is crucial to use multigrid solvers in combination with parallel computers. A number of approaches are currently in use in the community, which can broadly be divided into coupled and decoupled approaches. In the decoupled approach, an algebraic or geometric multigrid method is used as a preconditioner for the velocity equations only while an iterative approach such as Schur complement reduction used to solve the outer velocity-pressure equations. In the coupled approach, on the other hand, a multigrid approach is applied to both the velocity and pressure equations. The coupled multigrid approaches are typically employed in combination with staggered finite difference discretizations, whereas the decoupled approach is the method of choice in many of the existing finite element codes. Yet, it is unclear whether there are differences in speed between the two approaches, and if so, how this depends on the initial guess. Here, we implemented both approaches in combination with a staggered finite difference discretization and test the robustness of the two approaches with respect to large jumps in viscosity contrast, as well as their computational efficiency as a function of the initial guess. Acknowledgements. Funding was provided by the European Research Council under the European Community's Seventh Framework Program (FP7/2007-2013) / ERC Grant agreement #258830. Numerical computations have been performed on JUQUEEN of the Jülich high-performance computing center.

Kaus, Boris; Popov, Anton; Püsök, Adina

2014-05-01

55

Finite difference scheme for semiconductor Boltzmann equation with nonlinear collision operator

Directory of Open Access Journals (Sweden)

Full Text Available In this paper the Boltzmann equation describing electron flow ina semiconductor device is considered. The doping profile and the selfconsistent electric field are related by the Poisson equation. Time dependentsolutions to the Boltzmann-Poisson system are obtained byusing a finite difference numerical scheme. The collision operator ofthe Boltzmann equation models the scattering processes between electronsand phonons which are assumed in thermal equilibrium, and takesinto account the Pauli’s exclusion principle. Numerical results for onedimensional n+ ? n ? n+ silicon device are shown.

C. L. R. Milazzo

2013-01-01

56

Data transposition and helical scheme in prestack finite-difference migration

Energy Technology Data Exchange (ETDEWEB)

New wave equations are constantly being developed to respond to the increased demand for imaging complex geologic structures. In this study, the authors analyze the efficiency of prestack finite-difference migration with particular focus on data transposition in the two-way splitting algorithm. They presented a tiled data transposition algorithm which achieves good efficiency through greater access coherence. A helical scheme which eliminates the need for data transposition in common-shot migration was also presented. This newly developed algorithm speeds up downward extrapolation by 30 to 40 per cent and produces an image quality that is comparable to non-helical methods. 13 refs., 1 tab., 5 figs.

Zhang, Y. [Veritas DGC Inc., Houston, TX (United States); Liang, J. [Veritas DGC Inc., Calgary, AB (Canada); Zhang, G. [Chinese Academy of Sciences, Beijing (China)

2003-07-01

57

On the convergence of certain finite-difference schemes by an inverse-matrix method

The inverse-matrix method of analyzing the convergence of the solution of a given system of finite-difference equations to the solution of the corresponding system of partial-differential equations is discussed and generalized. The convergence properties of a time- and space-centered differencing of the diffusion equation are analyzed as well as a staggered grid differencing of the Cauchy-Riemann equations. These two schemes are significant since they serve as simplified model algorithms for two recently developed methods used to calculate nonlinear aerodynamic flows.

Steger, J. L.; Warming, R. F.

1975-01-01

58

A finite difference scheme for three-dimensional steady laminar incompressible flow

A finite difference scheme for three-dimensional steady laminar incompressible flows is presented. The Navier-Stokes equations are expressed conservatively in terms of velocity and pressure increments (delta form). First order upwind differences are used for first order partial derivatives of velocity increments resulting in a diagonally dominant matrix system. Central differences are applied to all other terms for second order accuracy. The SIMPLE pressure correction algorithm is used to satisfy the continuity equation. Numerical results are presented for cubic cavity flow problems for Reynolds numbers up to 2000 and are in good agreement with other numerical results.

Hwang, Danny P.; Huynh, Hung T.

1987-01-01

59

International Nuclear Information System (INIS)

The coarse mesh finite difference (CMFD) formulation is applied to the heterogeneous whole transport calculation as a means of efficient acceleration as well as simplified representation. The CMFD formulation enables dynamic homogenization of the cells during the iterative solution process such that the heterogeneous transport solution can be preserved. Dynamic group condensation is also possible with a two-level CMFD formulation involving alternate multi-group and two group calculations. The two-dimensional discrete ordinates (SN) method is used as the kernel to generate the heterogeneous solution and the CMFD solution provides the SN kernel with much faster converged fission and scattering source distributions. Through the applications to various problems including different compositions, sizes, and energy groups, it is shown that the performance of the CMFD formulation is superior in that the number of the SN Transport sweeps can be reduced to about 10 while reproducing exactly the original transport solution. (authors)

60

Energy Technology Data Exchange (ETDEWEB)

The coarse mesh finite difference (CMFD) formulation is applied to the heterogeneous whole transport calculation as a means of efficient acceleration as well as simplified representation. The CMFD formulation enables dynamic homogenization of the cells during the iterative solution process such that the heterogeneous transport solution can be preserved. Dynamic group condensation is also possible with a two-level CMFD formulation involving alternate multi-group and two group calculations. The two-dimensional discrete ordinates (SN) method is used as the kernel to generate the heterogeneous solution and the CMFD solution provides the SN kernel with much faster converged fission and scattering source distributions. Through the applications to various problems including different compositions, sizes, and energy groups, it is shown that the performance of the CMFD formulation is superior in that the number of the SN Transport sweeps can be reduced to about 10 while reproducing exactly the original transport solution. (authors)

Zhaopeng, Z.; Downar, T. J. [Purdue Univ., School of Nuclear Engineering, West Lafayette, IN 47907-1290 (United States); DeHart, M. D.; Clarno, K. T. [Oak Ridge National Laboratory, Nuclear Science and Technology Div., Oak Ridge, TN 37831-6370 (United States)

2006-07-01

61

A fourth-order finite difference scheme for the numerical solution of 1D linear hyperbolic equation

Directory of Open Access Journals (Sweden)

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.

Akbar Mohebbi

2013-10-01

62

A 3D staggered-grid finite difference scheme for poroelastic wave equation

Three dimensional numerical modeling has been a viable tool for understanding wave propagation in real media. The poroelastic media can better describe the phenomena of hydrocarbon reservoirs than acoustic and elastic media. However, the numerical modeling in 3D poroelastic media demands significantly more computational capacity, including both computational time and memory. In this paper, we present a 3D poroelastic staggered-grid finite difference (SFD) scheme. During the procedure, parallel computing is implemented to reduce the computational time. Parallelization is based on domain decomposition, and communication between processors is performed using message passing interface (MPI). Parallel analysis shows that the parallelized SFD scheme significantly improves the simulation efficiency and 3D decomposition in domain is the most efficient. We also analyze the numerical dispersion and stability condition of the 3D poroelastic SFD method. Numerical results show that the 3D numerical simulation can provide a real description of wave propagation.

Zhang, Yijie; Gao, Jinghuai

2014-10-01

63

Overlap grid is usually used in numerical simulation of flow with complex geometry by high order finite difference scheme. It is difficult to generate overlap grid and the connectivity information between adjacent blocks, especially when interpolation is required for non-coincident overlap grids. In this study, an interface flux reconstruction (IFR) method is proposed for numerical simulation using high order finite difference scheme with multi-block structured grids. In this method the neighboring blocks share a common face, and the fluxes on each block are matched to set the boundary conditions for each interior block. Therefore this method has the promise of allowing discontinuous grids on either side of an interior block interface. The proposed method is proven to be stable for 7-point central DRP scheme coupled with 4-point and 5-point boundary closure schemes, as well as the 4th order compact scheme coupled with 3rd order boundary closure scheme. Four problems are numerically solved with the developed code to validate the interface flux reconstruction method in this study. The IFR method coupled with the 4th order DRP scheme or compact scheme is validated to be 4th order accuracy with one and two dimensional waves propagation problems. Two dimensional pulse propagation in mean flow is computed with wavy mesh to demonstrate the ability of the proposed method for non-uniform grid. To demonstrate the ability of the proposed method for complex geometry, sound scattering by two cylinders is simulated and the numerical results are compared with the analytical data. It is shown that the numerical results agree well with the analytical data. Finally the IFR method is applied to simulate viscous flow pass a cylinder at Reynolds number 150 to show its capability for viscous problem. The computed pressure coefficient on the cylinder surface, the frequency of vortex shedding, the lift and drag coefficients are presented. The numerical results are compared with the data of other researchers, and a good agreement is obtained. The validations imply that the proposed IFR method is accurate and effective for inviscid and viscous problems with complex geometry.

Gao, Junhui

2013-05-01

64

Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights ?k and ?k, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r-1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators ?k in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ?, which is added to ?k to avoid a division by zero in the denominator of ?k. However, ? also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r-1), ? should be of ?(?x2) (?(?xm) means that ??C?xm for some C independent of ?x, as ?x?0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of ?k. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ?. Here, using same special properties of ?k, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ?=?(?xm), where m(r,p)?2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ? allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of ?k. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2?p?r-1, at the cost of increased numerical dissipation. Compact formulas of ?k for WENO schemes are also presented.

Don, Wai-Sun; Borges, Rafael

2013-10-01

65

A finite difference scheme for magneto-thermo analysis of an infinite cylinder

Directory of Open Access Journals (Sweden)

Full Text Available A finite different scheme as well as least-square method is presented for the magneto-thermo analysis of an infinite functionally graded hollow cylinder. The radial displacement, mechanical stresses and temperature as well as the electromagnetic stress are investigated along the radial direction of the cylinder. Material properties are assumed to be graded in the radial direction according to a novel exponential-law distribution in terms of the volume fractions of the metal and ceramic constituents. The governing second-order differential equations are derived from the equations of motion and the heat-conduction equation. The system of differential equations is solved numerically and some plots for displacement, radial stress, and temperature are presented.

Daoud S. Mashat

2010-11-01

66

DEFF Research Database (Denmark)

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) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the nonlinear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of 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 insight into into the numerical behavior of this rather complicated system of nonlinear PDEs.

Bingham, Harry B.; Madsen, Per A.

2004-01-01

67

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the pre-processing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.

Lombard, B.; Piraux, J.; Gélis, C.; Virieux, J.

2008-01-01

68

A dynamic grid point allocation scheme for the Characteristic Finite Difference Method

International Nuclear Information System (INIS)

Codes for predicting the events which can occur during a reactor loss-of-coolant accident (LOCA) must be able to cope with the essentially steady flows of normal operation and long-term cooling, as well as with the very severe transients occurring during the blowdown and water injection phases. Although large timesteps and a coarse grid are adequate for steady flows, smaller timesteps and a finer grid are necessary to preserve accuracy during the fast transients. The RAMA code, based on the characteristic finite difference method, has a dynamic grid point allocation scheme which helps to control spatial truncation error. In this scheme, grid points are inserted in regions where large spatial gradients occur in the flow-boiling variables and are deleted when these gradients subside. As well, the mesh of grid points is free to move with the gradients. The algorithm used is similar to that for dynamic timestep control, where large temporal gradients in the flow-boiling variables dictate small timesteps and larger timesteps are used for shallower gradients. The goal is to maintain a prescribed accuracy in the calculation during the entire LOCA prediction while using the minimum number of grid points at each step and, in total, the minimum number of timesteps. Examples are presented which demonstrate the computational efficiency of the scheme, including a benchmark problem involving cold water injection and an experimental case involving both blowdown and cold water injection phases

69

International Nuclear Information System (INIS)

An entirely new discrete levels segment 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 segment 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 as gamma-ray energy and branching percentage. Isomer flags for half-lives longer than 1 s have been introduced. For those 1,277 nuclei having at least ten known levels the cutoff level numbers Nm have been determined from fits to the cumulative number of levels. The level numbers Nc associated with the cutoff energies Uc, corresponding to the upper energy limit of levels with unique spin and parity, have been included for each nuclide. The segment has the form of an ASCII file which follows the extended ENEA Bologna convention. For the RIPL Starter File the new Budapest file is recommended as a Discrete Level Schemes Segment because it is most complete, up-to-date, and also well documented. Moreover, the cutoff energies have been determined in a consistent way, giving also hints about basic level density parameters. The recommended files are budapest-levels.dat and budapest-cumulative.dat. As alternative choices, the libraries from Beijing, Bologna, JAERI, Obninsk and Livermore may also be used for special applications. (author)

70

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 l2 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 stability of the whole space-time scheme. An exactly preserved functional for the norm of the Dirac spinor on the staggered grid is presented. Simulations of Gaussian wave packets, leaving the computational domain without reflection, demonstrate the quality of the DTBCs numerically, as well as the importance of a faithful representation of the energy-momentum dispersion relation on a grid.

Hammer, René; Pötz, Walter; Arnold, Anton

2014-01-01

71

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.

Mickens, Ronald E.

1996-01-01

72

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...

Hammer, René; Arnold, Anton

2013-01-01

73

Landing-gear noise prediction using high-order finite difference schemes

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.

Liu, Wen; Wook Kim, Jae; Zhang, Xin; Angland, David; Caruelle, Bastien

2013-07-01

74

This paper deals with stability in the numerical solution of general one-dimensional partial differential equations with variable coefficients. We will generalize stability results for central finite difference schemes on non-uniform grids that were obtained by In't Hout & Volders (2009) for the Black-Scholes equation. Subsequently we will apply our stability results to the CEV model.

Volders, Kim

2010-09-01

75

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.

Martowicz, A.; Ruzzene, M.; Staszewski, W. J.; Rimoli, J. J.; Uhl, T.

2014-03-01

76

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.

Mickens, Ronald E.

1989-01-01

77

We present a finite-difference scheme which solves the Stokes problem in the presence of curvilinear non-conforming interfaces and provides second-order accuracy on physical field (velocity, vorticity) and especially on pressure. The gist of our method is to rely on the Helmholtz decomposition of the Stokes equation: the pressure problem is then written in an integral form devoid of the spurious sources known to be the cause of numerical boundary layer error in most implementations, leading to a discretization which guarantees a strict enforcement of mass conservation. The ghost method is furthermore used to implement the boundary values of pressure and vorticity near curved interfaces.

Hammouti, Abdelkader

2011-01-01

78

This paper concerns the numerical solution of the Black-Scholes PDE with a Neumann boundary condition on the right boundary. We consider finite difference schemes for the semi-discretization, which leads to a system of ODEs with corresponding matrix M. In this paper stability bounds for exp(tM) (t ? 0) are proved. A scaled version of the Euclidean norm, denoted by ? ? ?H is considered. The advection and diffusion term of the PDE are analyzed separately. It turns out that the Neumann boundary condition leads to a growth of ?exp(tM)?H with the number of grid points m for the pure advection problem.

Volders, K.

2012-09-01

79

Digital Repository Infrastructure Vision for European Research (DRIVER)

This technical report yields detailed calculations of the paper [1] (B. Bid\\'egaray-Fesquet, "Stability of FD-TD schemes for Maxwell-Debye and Maxwell-Lorentz equations", Technical report, LMC-IMAG, 2005) which have been however automated since (see http://ljk.imag.fr/membres/Brigitte.Bidegaray/NAUtil/). It deals with the stability analysis of various finite difference schemes for Maxwell--Debye and Maxwell--Lorentz equations. This work gives a systematic and rigorous con...

Bide?garay-fesquet, Brigitte

2008-01-01

80

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.

Fisher, Travis C.; Carpenter, Mark H.; Nordstroem, Jan; Yamaleev, Nail K.; Swanson, R. Charles

2011-01-01

81

Energy Technology Data Exchange (ETDEWEB)

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a ''rough'' coefficient function k(x). we show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general L{sup p} compactness criterion. (author)

Karlsen, Kenneth Hvistendal; Risebro, Nils Henrik

2000-09-01

82

A few families of counterexamples are provided to "A proof that the discrete singular convolution (DSC)/Lagrange-distributed approximation function (LDAF) method is inferior to high order finite differences", Journal of Computational Physics, 214, 538-549 (2006).

Wei, G W; Zhao, Shan

2006-01-01

83

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.

Surkova, Maryna; Pavlenko, Yvan; Van Oost, Guido; Van Eester, Dirk; De Zutter, Daniël

2014-05-01

84

In Xu (2013) [14], a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work (Xu, 2013) [14], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in Levy (2005) [3]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.

Xiong, Tao; Qiu, Jing-Mei; Xu, Zhengfu

2013-11-01

85

For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion.

Tan, Sirui; Huang, Lianjie

2014-11-01

86

Invariant discretization schemes for the shallow-water equations

Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions using difference invariants. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e. the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy and mass is evaluated for the invariant and the non-invariant...

Bihlo, Alexander

2012-01-01

87

On the third-order upwind finite-difference scheme in numerical analysis for initial-value problems

International Nuclear Information System (INIS)

The purpose of this paper is to present highly accurate and stable numerical solutions for the time-dependent Navier-Stokes equations. This numerical method utilizes uniformly third-order finite-difference scheme based on Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms(QUICKEST) combined with the Marker and Cell computing technique(QUICKESTMAC) for unsteady convection-dominated flows. The solutions for a time-marching square cavity flow are obtained with Reynolds number 10,000 and the computed results prove the validity of this QUICKESTMAC method. (author)

88

A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene

Energy Technology Data Exchange (ETDEWEB)

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.

Brinkman, D., E-mail: Daniel.Brinkman@asu.edu [Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom); School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287 (United States); Heitzinger, C., E-mail: Clemens.Heitzinger@asu.edu [Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom); AIT Austrian Institute of Technology, A-1220 Vienna (Austria); School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287 (United States); Markowich, P.A., E-mail: Peter.Markowich@kaust.edu.sa [Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900 (Saudi Arabia); Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

2014-01-15

89

A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene

International Nuclear Information System (INIS)

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

90

A convergent 2D finite-difference scheme for the Dirac-Poisson system and the simulation of graphene

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.

Brinkman, D.; Heitzinger, C.; Markowich, P. A.

2014-01-01

91

Numerical pricing of options using high-order compact finite difference schemes

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.

Tangman, D. Y.; Gopaul, A.; Bhuruth, M.

2008-09-01

92

The convergence and accuracy of three finite-difference solution schemes (SIMPLE, SIMPLEC and PISO) are tested and evaluated for steady, incompressible, and two-dimensional flows in nonorthogonal body-fitted coordinate systems. When representing the governing equations in a generalized coordinate system, complex cross-derivative pressure terms that involve nodal values on 'corner points' appear. To maximize computational efficiency with a five-point matrix solver, the corner points are generally omitted. However, the neglect of these terms may lead to deterioration in the convergence rate for fairly skewed grids and divergence for highly skewed grids. It is shown in this paper that the PISO solution scheme, with its multicorrector steps, allows the inclusion of the cross-derivative pressure terms and offers a distinct advantage in computational efficiency over the commonly used SIMPLE and SIMPLEC methods.

Hadid, A. H.; Chan, D. C.; Issa, R. I.; Sindir, M. M.

1988-07-01

93

In this work, we present two numerical methods to approximate solutions of systems of dissipative sine-Gordon equations that arise in the study of one-dimensional, semi-infinite arrays of Josephson junctions coupled through superconducting wires. Also, we present schemes for the total energy of such systems in association with the finite-difference schemes used to approximate the solutions. The proposed methods are conditionally stable techniques that yield consistent approximations not only in the domains of the solution and the total energy, but also in the approximation to the rate of change of energy with respect to time. The methods are employed in the estimation of the threshold at which nonlinear supratransmission takes place, in the presence of parameters such as internal and external damping, generalized mass, and generalized Josephson current. Our results are qualitatively in agreement with the corresponding problem in mechanical chains of coupled oscillators, under the presence of the same paramete...

Macías-Díaz, J E; 10.1016/j.cnsns.2009.04.017

2011-01-01

94

This technical report yields detailed calculations of the paper [1] (B. Bid\\'egaray-Fesquet, "Stability of FD-TD schemes for Maxwell-Debye and Maxwell-Lorentz equations", Technical report, LMC-IMAG, 2005) which have been however automated since (see http://ljk.imag.fr/membres/Brigitte.Bidegaray/NAUtil/). It deals with the stability analysis of various finite difference schemes for Maxwell--Debye and Maxwell--Lorentz equations. This work gives a systematic and rigorous continuation to Petropoulos previous work [5] (P.G. Petropoulos.,"Stability and phase error analysis of FD-TD in dispersive dielectrics", IEEE Transactions on Antennas and Propagation, 42(1):62--69, 1994).

Bidégaray-Fesquet, Brigitte

2008-01-01

95

An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.

Harten, A.; Tal-Ezer, H.

1981-01-01

96

Invariant Discretization Schemes Using Evolution-Projection Techniques

Directory of Open Access Journals (Sweden)

Full Text Available Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.

Alexander Bihlo

2013-08-01

97

A family of energy stable, skew-symmetric finite difference schemes on collocated grids

A simple scheme for incompressible, constant density flows is presented, which avoids odd-even decoupling for the Laplacian on a collocated grids. Energy stability is implied by maintaining strict energy conservation. Momentum is conserved. Arbitrary order in space and time can easily be obtained. The conservation properties hold on transformed grids.

Reiss, Julius

2014-01-01

98

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: English Abstract in english We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced- [...] order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.

Carlos, Duque-Daza; Duncan, Lockerby; Carlos, Galeano.

2011-12-01

99

The paper considers the leap-frog finite-difference method (Kreiss and Oliger, 1973) for systems of partial differential equations of the form du/dt = dF/dx + dG/dy + dH/dz, where d denotes partial derivative, u is a q-component vector and a function of x, y, z, and t, and the vectors F, G, and H are functions of u only. The original leap-frog algorithm is shown to admit a modification that improves on the stability conditions for two and three dimensions by factors of 2 and 2.8, respectively, thereby permitting larger time steps. The scheme for three dimensions is considered optimal in the sense that it combines simple averaging and large time steps.

Abarbanel, S.; Gottlieb, D.

1976-01-01

100

This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.

Harten, A.; Tal-Ezer, H.

1981-01-01

101

DEFF Research Database (Denmark)

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) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the non-linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of 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 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 non-linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non-linear PDEs.

Fuhrman, David R.; Bingham, Harry B.

2004-01-01

102

An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation

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.

Zhan, Ge; Pestana, Reynam C.; Stoffa, Paul L.

2013-04-01

103

In acoustic logging-while-drilling (ALWD) finite difference in time domain (FDTD) simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns (radius ˜27 mm). Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole (the difference is >30 times), the stability and efficiency of the perfectly matched layer (PML) scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain (FDTD) in the ALWD simulation, including field-splitting PML (SPML), multiaxial PML(MPML), non-splitting PML (NPML), and complex frequency-shifted PML (CFS-PML). The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one d 0. The optimal parameter space for the maximum value of the linear frequency-shifted factor ( ? 0) and the scaling factor ( ? 0) depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to PML parameters, and the error will decrease as the PML thickness increases.

Wang, Hua; Tao, Guo; Shang, Xue-Feng; Fang, Xin-Ding; Burns, Daniel R.

2013-12-01

104

A rigorous full-vector analysis based on the finite-difference mode-matching method is presented for three-dimensional optical wave propagation problems. The computation model is facilitated by a perfectly matched layer (PML) terminated with a perfectly reflecting boundary condition (PRB). The complex modes including both the guided and the radiation fields of the three-dimensional waveguide with arbitrary index profiles are computed by a finite-difference scheme. The method is applied to and validated by the analysis of the facet reflectivity of a buried waveguide and the power exchange of a periodically loaded dielectric waveguide polarization rotator. PMID:18958093

Mu, Jianwei; Huang, Wei-Ping

2008-10-27

105

Compact Finite Difference Approximations for Space Fractional Diffusion Equations

Based on the weighted and shifted Gr\\"{u}nwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the one and two dimensional space fractional diffusion equations. The detailed numerical stability and error analysis are theoretically performed. We theoretically prove and numerically verify that the provided numerical schemes have the convergent orders 3 in space and 2 in time.

Zhou, Han; Deng, Weihua

2012-01-01

106

International Nuclear Information System (INIS)

In this paper we suggest the linear scheme for the three-dimensional neutron diffusion equation approximation. In this scheme we use the finite element method with bi-quadratic test functions for x,y approximation. For z approximation of the equation we use the finite difference method. Theoretically, such a scheme provides convergence with a high order of accuracy: the third order or higher for x,y variables and the second order for z variable. The scheme provides simulations with space grid refinement. Our computational investigations showed that accuracy of calculations is acceptable even at the base (the coarsest) grid. It provides significant reduction of calculation time compared to simulations based on the second accuracy order schemes. Our scheme was implemented in the KORAT 3D code. The RBMK reactor was simulated as the test problem with a different detailed order. The results of the computational investigation of convergence were compared with the results obtained by the second accuracy order scheme. (authors)

107

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.

Wieselquist, William A.; Anistratov, Dmitriy Y.; Morel, Jim E.

2014-09-01

108

International Nuclear Information System (INIS)

In this paper, we use the staggered grid, the auxiliary grid, the rotated staggered grid and the non-staggered grid finite-difference methods to simulate the wavefield propagation in 2D elastic tilted transversely isotropic (TTI) and viscoelastic TTI media, respectively. Under the stability conditions, we choose different spatial and temporal intervals to get wavefront snapshots and synthetic seismograms to compare the four algorithms in terms of computational accuracy, CPU time, phase shift, frequency dispersion and amplitude preservation. The numerical results show that: (1) the rotated staggered grid scheme has the least memory cost and the fastest running speed; (2) the non-staggered grid scheme has the highest computational accuracy and least phase shift; (3) the staggered grid has less frequency dispersion even when the spatial interval becomes larger. (paper)

109

In modern magnetic resonance imaging (MRI), there are concerns for the health and safety of patients and workers repeatedly exposed to magnetic fields, and therefore accurate and efficient evaluation of in situ electromagnetic field (EMF) distributions has gained a lot of significance. This paper presents a Biconjugate Gradient Method (BiCG) to efficiently implement the quasi-static finite-difference scheme (QSFD), which has been widely utilized to model and analyze magnetically induced electric fields and currents within the human body during the operation of the MRI systems and in other settings. The proposed BiCG method shows computational advantages over the iterative, successive over-relaxation (SOR) algorithm. The scheme has been validated against other known solutions on a lossy, multilayered ellipsoid phantom excited by an ideal loop coil. Numerical results on a 3D human body model demonstrate that the convergence time and memory consumption is significantly reduced using the BiCG method. PMID:18001995

Wang, H; Liu, F; Trakic, A; Xia, L; Crozier, S

2007-01-01

110

A fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature.

Tian, Zhenfu; Ge, Yongbin

2003-02-01

111

Energy Technology Data Exchange (ETDEWEB)

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)

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

112

Diffusion acceleration schemes in R-Z discrete ordinates scheme

International Nuclear Information System (INIS)

The discrete ordinates method is an accurate method for analysing the criticality and flux in a nuclear reactor. TRITAC is a x-y-z discrete ordinates code for solving the S-N equations for multigroup criticality calculations. In addition to the initial diffusion calculation for flux guess, it uses DSA (diffusion synthetic acceleration) scheme for improving the efficiency of inner and outer iterations. Recently, the r-z version of this method has been developed. For incorporating the diffusion synthetic acceleration scheme in r-z geometry, S-N diffusion theory in r-z program is used. The main difference between S-N and standard diffusion theory is the radial dependence of the diffusion coefficient in the S-N diffusion theory. In this paper, the r-z program to analyse the criticality of a light water reactor core which is surrounded by layers of water and steel is used. It is concluded that a good initial diffusion theory guess is more important for accelerating the S-N iterations that the DSA scheme. Further, the S-N diffusion theory is shown to be less effective than the standard diffusion theory. (author). 4 refs., 2 tabs

113

Mimetic finite difference method

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.

Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail

2014-01-01

114

Explicit finite difference methods for the delay pseudoparabolic equations.

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

Amirali, I; Amiraliyev, G M; Cakir, M; Cimen, E

2014-01-01

115

A Comparison of Continuous Mass-lumped Finite Elements and Finite Differences for 3D:

Digital Repository Infrastructure Vision for European Research (DRIVER)

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 geometries near sharp interfaces. We compared the fourth-order finite-difference method to fourth-order continuous masslumped finite elements in terms of accuracy and computational cost. The results s...

Zhebel, E.; Minisini, S.; Kononov, A.; Mulder, W. A.

2012-01-01

116

Steady discrete shocks of high-order RBC schemes

Exact expressions of steady discrete shocks are found for a class of dissipative compact schemes approximating a one-dimensional nonlinear hyperbolic problem with a 3rd, 5th and 7th order of accuracy. A discrete solution is given explicitly for the inviscid Bürgers equation and the oscillatory nature of the shock profiles is determined according to the scheme order and to the shock location with respect to the mesh.

Lerat, Alain

2013-11-01

117

We investigate the connections between several recent methods for the discretization of anisotropic 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 be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

Droniou, Jerome; Gallouët, Thierry; Herbin, Raphaele

2008-01-01

118

Finite difference methods for the Infinity Laplace and p-Laplace equations

Digital Repository Infrastructure Vision for European Research (DRIVER)

We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosity solution of the underlying equation. We build a semi-implicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of iterations is independen...

Oberman, Adam M.

2011-01-01

119

The accuracy of finite difference methods is related to the mesh choice and cell size. Concerning the micromagnetism of nano-objects, we show here that discretization issues can drastically affect the symmetry of the problem and therefore the resulting computed properties of lattices of interacting curved nanomagnets. In this paper, we detail these effects for the multi-axis kagome lattice. Using the OOMMF finite difference method, we propose an alternative way of discretizing the nanomagnet shape via a variable moment per cell scheme. This method is shown to be efficient in reducing discretization effects. PMID:23803392

Riahi, H; Montaigne, F; Rougemaille, N; Canals, B; Lacour, D; Hehn, M

2013-07-24

120

International Nuclear Information System (INIS)

The accuracy of finite difference methods is related to the mesh choice and cell size. Concerning the micromagnetism of nano-objects, we show here that discretization issues can drastically affect the symmetry of the problem and therefore the resulting computed properties of lattices of interacting curved nanomagnets. In this paper, we detail these effects for the multi-axis kagome lattice. Using the OOMMF finite difference method, we propose an alternative way of discretizing the nanomagnet shape via a variable moment per cell scheme. This method is shown to be efficient in reducing discretization effects. (paper)

121

Optimized Discretization Schemes For Brain Images

Digital Repository Infrastructure Vision for European Research (DRIVER)

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 TV...

USHA RANI.N,; Subbaiah, Dr P. V.; Venkata Rao, Dr D.

2011-01-01

122

On discrete functional inequalities for some finite volume schemes

Digital Repository Infrastructure Vision for European Research (DRIVER)

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(\\Omega)$ into $L^{N/(N-1)}(\\Omega)$ for a Lipschitz domain $ \\Omega \\subset \\mathbb{R}^{N}$, with $N \\geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approxim...

Bessemoulin-chatard, Marianne; Chainais-hillairet, Claire; Filbet, Francis

2012-01-01

123

On the accuracy and efficiency of finite difference solutions for nonlinear waves

DEFF Research Database (Denmark)

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-uniform grid. Time-integration is performed using a fourth-order Runge-Kutta scheme. The linear accuracy, stability and convergence properties of the method are analyzed in two-dimensions, and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as providing an optimal balance between accuracy and complexity for practical purposes.

Bingham, Harry B.

2006-01-01

124

Optimized Discretization Schemes For Brain Images

Directory of Open Access Journals (Sweden)

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.

USHA RANI.N,

2011-02-01

125

Mimetic finite difference methods in image processing

Scientific Electronic Library Online (English)

Full Text Available SciELO Brazil | Language: English Abstract in english We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and res [...] toration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference). We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.

C., Bazan; M., Abouali; J., Castillo; P., Blomgren.

126

Mimetic finite difference methods in image processing

Directory of Open Access Journals (Sweden)

Full Text Available We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and restoration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference. We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.

C. Bazan

2011-01-01

127

Discrete-time interacting quantum walks and quantum Hash schemes

Through introducing discrete-time quantum walks on the infinite line and on circles, we present a kind of two-particle interacting quantum walk which has two kinds of interactions. We investigate the characteristics of this kind of quantum walk and the time evolution of the two particles. Then we put forward a kind of quantum Hash scheme based on two-particle interacting quantum walks and discuss their feasibility and security. The security of this kind of quantum Hash scheme relies on the infinite possibilities of the initial state rather than the algorithmic complexity of hard problems, which will greatly enhance the security of the Hash schemes.

Li, Dan; Zhang, Jie; Guo, Fen-Zhuo; Huang, Wei; Wen, Qiao-Yan; Chen, Hui

2013-03-01

128

Parallel discrete-event simulation schemes with heterogeneous processing elements

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

Kim, Yup; Kwon, Ikhyun; Chae, Huiseung; Yook, Soon-Hyung

2014-07-01

129

Parallel discrete-event simulation schemes with heterogeneous processing elements.

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind. PMID:25122349

Kim, Yup; Kwon, Ikhyun; Chae, Huiseung; Yook, Soon-Hyung

2014-07-01

130

The coupled double diffusive natural convection and radiation in a tilted and differentially heated square cavity containing a non-gray air-CO2 (or air-H2O) mixtures was numerically investigated. The horizontal walls are insulated and impermeable and the vertical walls are maintained at different temperatures and concentrations. The hybrid lattice Boltzmann method with the multiple-relaxation time model is used to compute the hydrodynamics and the finite difference method to determine temperatures and concentrations. The discrete ordinates method combined to the spectral line-based weighted sum of gray gases model is used to compute the radiative term and its spectral aspect. The effects of the inclination angle on the flow, thermal and concentration fields are analyzed for both aiding and opposing cases. It was found that radiation gas modifies the structure of the velocity and thermal fields by generating inclined stratifications and promoting the instabilities in opposing flows.

Moufekkir, Fayçal; Moussaoui, Mohammed Amine; Mezrhab, Ahmed; Naji, Hassan

2014-09-01

131

High-resolution finite-difference algorithms for conservation laws

International Nuclear Information System (INIS)

A new class of Total Variation Decreasing (TVD) schemes for 2-dimensional scalar conservation laws is constructed using either flux-limited or slope-limited numerical fluxes. The schemes are proven to have formal second-order accuracy in regions where neither u/sub x/ nor y/sub y/ vanishes. A new class of high-resolution large-time-step TVD schemes is constructed by adding flux-limited correction terms to the first-order accurate large-time-step version of the Engquist-Osher scheme. The use of the transport-collapse operator in place of the exact solution operator for the construction of difference schemes is studied. The production of spurious extrema by difference schemes is studied. A simple condition guaranteeing the nonproduction of spurious extrema is derived. A sufficient class of entropy inequalities for a conservation law with a flux having a single inflection point is presented. Finite-difference schemes satisfying a discrete version of each entropy inequality are only first-order accurate

132

Finite-difference models of ordinary differential equations - Influence of denominator functions

This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.

Mickens, Ronald E.; Smith, Arthur

1990-01-01

133

Novel discretization schemes for the numerical simulation of membrane dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements. Rather than time-marching, the entire space-time domain was discretized and solved simultaneously. Second-order rates of convergence in both space and time were observed in numerical studies. Slight high-frequency noise was filtered during post-processing. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. The observed rate of convergence was two for both displacement and strain. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate.

Kolsti, Kyle F.

134

We consider an initial-boundary value problem for a generalized 2D time-dependent Schr\\"odinger equation on a semi-infinite strip. For the Crank-Nicolson finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the related practical error analysis.

Ducomet, Bernard; Zlotnik, Ilya

2013-01-01

135

On Cryptographic Schemes Based on Discrete Logarithms and Factoring

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.

Joye, Marc

136

The Relation of Finite Element and Finite Difference Methods

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Vinokur, M.

1976-01-01

137

High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves

DEFF Research Database (Denmark)

The incompressible Euler equations are solved with a free surface, the position of which is captured by applying an Eulerian kinematic boundary condition. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied implicitly, at the end of each time stage, by constructing the pressure from a discrete Poisson equation, derived from the discrete continuity and momentum equations and taking the time-dependent physical domain into account. An efficient preconditionedDefect Correction (DC) solution of the discrete Poisson equation for the pressure is presented, in which the preconditioning step is based on an order-multigrid formulation with a direct solution on the lowest order-level. This ensures fast convergence of the DC method with a computational effort which scales linearly with the problem size. Results obtained 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.

Christiansen, Torben Robert Bilgrav; Bingham, Harry B.

2012-01-01

138

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

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.

Kouatchou, Jules

1999-01-01

139

Simulation of axi-symmetric flow towards wells: A finite-difference approach

A detailed finite-difference approach is presented for the simulation of transient radial flow in multi-layer systems. The proposed discretization scheme simulates drawdown within the well more accurately than commonly applied schemes. The solution is compared to existing (semi) analytical models for the simulation of slug tests and pumping tests with constant discharge in single- and multi-layer systems. For all cases, it is concluded that the finite-difference model approximates drawdown to acceptable accuracy. The main advantage of finite-difference approaches is the ability to account for the varying saturated thickness in unconfined top layers. Additionally, it is straightforward to include radial variation of hydraulic parameters, which is useful to simulate the effect of a finite-thickness well skin. Aquifer tests with variable pumping rate and/or multiple wells may be simulated by superposition. The finite-difference solution is implemented in MAxSym, a MATLAB tool which is designed specifically to simulate axi-symmetric flow. Alternatively, the presented equations can be solved using a standard finite-difference model. A procedure is outlined to apply the same approach with MODFLOW. The required modifications to the input parameters are much larger for MODFLOW than for MAxSym, but the results are virtually identical. The presented finite-difference solution may be used, for example, as a forward model in parameter estimation algorithms. Since it is applicable to multi-layer systems, its use is not limited to the simulation of traditional pumping and slug tests, but also includes advanced aquifer tests, such as multiple pumping tests or multi-level slug tests.

Louwyck, Andy; Vandenbohede, Alexander; Bakker, Mark; Lebbe, Luc

2012-07-01

140

Directory of Open Access Journals (Sweden)

Full Text Available The non-hydrostatic (NH compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D slice framework employing a spectral element method (SEM for the horizontal discretization and a finite difference method (FDM for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.

S.-J. Choi

2014-06-01

141

The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.

Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.

2014-06-01

142

The computer algebra approach of the finite difference methods for PDEs

International Nuclear Information System (INIS)

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

143

To stably employ multiband k.p model for analyzing the band structure in semiconductor heterostructures without spurious solutions (SSs), the Hermitian forward and backward difference (HFBD) scheme for finite difference method (FDM) is presented. The HFBD is the discretization scheme that eliminates the difference instability and employs the Burt-Foreman Hermitian operator ordering without geometric asymmetry. The difference instability arises from employing Foreman's strategy (FS). FS removes SSs caused by unphysical bowing in bulk dispersion curve meanwhile the HFBD is the only difference scheme that can accurately adapt for it. In comparison with other recent strategies, the proposed method in this paper is as accurate and reliable as FS, along with preserving the rapidness and simplicity of FDM. This difference scheme shows stable convergence without any SSs under variable grid size. Therefore, a wide range of experiment-determined band parameters can be applied to large-scale stable simulation with this method regardless of the SSs they originally generate.

Jiang, Yu; Ma, Xunpeng; Xu, Yun; Song, Guofeng

2014-11-01

144

Analysis of Compatible Discrete Operator Schemes for Elliptic Problems on Polyhedral Meshes

Digital Repository Infrastructure Vision for European Research (DRIVER)

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite an...

Bonelle, Je?ro?me; Ern, Alexandre

2014-01-01

145

A mimetic finite difference method for the Stokes problem with elected edge bubbles

Energy Technology Data Exchange (ETDEWEB)

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.

Lipnikov, K [Los Alamos National Laboratory; Berirao, L [DIPARTMENTO DI MATERMATICA

2009-01-01

146

Fully discrete high-order TVD schemes for a scalar hyperbolic conservation law

Digital Repository Infrastructure Vision for European Research (DRIVER)

In this paper we investigate fully discrete high-order TVD schemes for a scalar hyper- bolic conservation law using flux limiters . Formulae which define Courant number dependent TVD regions for second and third-order TVD schemes are established. A semi-empirical TVD procedure for an m-th order scheme (m ? 4) are proposed and tested.

Shi, Jian; Toro, E. F.

1993-01-01

147

Plane-wave responses of vertically heterogeneous structure models (1-D media) are often computed in seismology. For horizontally layered media, they can be calculated by semi-analytical methods such as the propagator matrix method. However, for the gradient velocity or randomly heterogeneous structures, we have to use numerical methods such as the finite-difference method. The conventional codes for the 2-D or 3-D finite-difference method require huge computer memory and long computation time even for calculating plane-wave responses of 1-D media. In this study we propose an efficient procedure to calculate plane-wave responses of arbitrary 1-D media using the finite-difference method. We first derive an elastodynamic equation of plane-wave incidence problem for vertically heterogeneous media by applying the Snell's law to 3-D elastodynamic equation. We then discretize the velocity-stress formulation of the derived elastodynamic equation using a staggered-grid finite-difference scheme of fourth-order accurate in space and second-order accurate in time. We also investigate the implement of the stress-free surface condition for the scheme, and perform a stability check of the total scheme through actual computations using a Fortran code based on it. We computed plane-wave responses of structure model with velocity gradient using the derived finite-difference method. We focused on the PS-converted phase and found a ``offset" phase appearing between the PS-converted phases generated at the top and bottom boundaries of a velocity-gradient layer on the surface responses of a structure model with velocity gradient due to a P-wave incidence. This phase can be emphasized by calculating the receiver function from the radial and vertical waveforms. In this study we also investigate the ``offset" phase attributed to the velocity gradient by numerical computations using the derived finite-difference method.

Tanaka, H.; Takenaka, H.; Nakamura, T.; Murakoshi, T.

2002-12-01

148

Using the Finite Difference Calculus to Sum Powers of Integers.

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.…

Zia, Lee

1991-01-01

149

On the Krylov maximum principle for discrete parabolic schemes

Directory of Open Access Journals (Sweden)

Full Text Available In previous works, we have established discrete versions of the Krylov maximum principle for parabolic operators, on general meshes in Euclidean space. In this article, we prove a variant of these estimates in terms of a discrete analogue of the determinant of the coefficient matrix in the differential operator case. Our treatment adapts key ideas from our previous work on the corresponding discrete Aleksandrov maximum principle in the elliptic case.

Hung-Ju Kuo

2009-12-01

150

The effect of artificial diffusion on discrete shock structures is examined for a family of schemes which includes scalar diffusion, convective upwind and split pressure (CUSP) schemes, and upwind schemes with characteristics splitting. The analysis leads to conditions on the diffusive flux such that stationary discrete shocks can contain a single interior point. The simplest formulation which meets these conditions is a CUSP scheme in which the coefficients of the pressure differences is fully determined by the coefficient of convective diffusion. It is also shown how both the characteristic and CUSP schemes can be modified to preserve constant stagnation enthalpy in steady flow, leading to four variants, the E and H-characteristic schemes, and the E and H-CUSP schemes. Numerical results are presented which confirm the properties of these schemes.

Jameson, Antony

1994-01-01

151

A New Signature Scheme Based on Factoring and Discrete Logarithm Problems

Directory of Open Access Journals (Sweden)

Full Text Available In 1994, He and Kiesler proposed a digital signature scheme which was based on the factoring and the discrete logarithm problem both. Same year, Shimin-Wei modi?ed the He-Kiesler signature scheme. In this paper, we propose an improvement of Shimin-Wei signature scheme based on factorization and discrete logarithm problem both with di?erent parameters and using a collision-free one-way hash function. In our opinion, our scheme is more secure than the earlier one.

Swati Verma

2012-07-01

152

An initial-boundary value problem for the $n$-dimensional ($n\\geq 2$) time-dependent Schr\\"odinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a P\\"{o}schl-Teller-like potential-barrier and a rectangular potential-well are also included.

Ducomet, Bernard; Romanova, Alla

2013-01-01

153

Implicit time-dependent finite different algorithm for quench simulation

Energy Technology Data Exchange (ETDEWEB)

A magnet in a fusion machine has many difficulties in its application because of requirement of a large operating current, high operating field and high breakdown voltage. A cable-in-conduit (CIC) conductor is the best candidate to overcome these difficulties. However, there remained uncertainty in a quench event in the cable-in-conduit conductor because of a difficulty to analyze a fluid dynamics equation. Several scientists, then, developed the numerical code for the quench simulation. However, most of them were based on an explicit time-dependent finite difference scheme. In this scheme, a discrete time increment is strictly restricted by CFL (Courant-Friedrichs-Lewy) condition. Therefore, long CPU time was consumed for the quench simulation. Authors, then, developed a new quench simulation code, POCHI1, which is based on an implicit time dependent scheme. In POCHI1, the fluid dynamics equation is linearlized according to a procedure applied by Beam and Warming and then, a tridiagonal system can be offered. Therefore, no iteration is necessary to solve the fluid dynamics equation. This leads great reduction of the CPU time. Also, POCHI1 can cope with non-linear boundary condition. In this study, comparison with experimental results was carried out. The normal zone propagation behavior was investigated in two samples of CIC conductors which had different hydraulic diameters. The measured and simulated normal zone propagation length showed relatively good agreement. However, the behavior of the normal voltage shows a little disagreement. These results indicate necessity to improve the treatment of the heat transfer coefficient in the turbulent flow region and the electric resistivity of the copper stabilizer in high temperature and high field region. (author).

Koizumi, Norikiyo; Takahashi, Yoshikazu; Tsuji, Hiroshi [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment

1994-12-01

154

Implicit time-dependent finite different algorithm for quench simulation

International Nuclear Information System (INIS)

A magnet in a fusion machine has many difficulties in its application because of requirement of a large operating current, high operating field and high breakdown voltage. A cable-in-conduit (CIC) conductor is the best candidate to overcome these difficulties. However, there remained uncertainty in a quench event in the cable-in-conduit conductor because of a difficulty to analyze a fluid dynamics equation. Several scientists, then, developed the numerical code for the quench simulation. However, most of them were based on an explicit time-dependent finite difference scheme. In this scheme, a discrete time increment is strictly restricted by CFL (Courant-Friedrichs-Lewy) condition. Therefore, long CPU time was consumed for the quench simulation. Authors, then, developed a new quench simulation code, POCHI1, which is based on an implicit time dependent scheme. In POCHI1, the fluid dynamics equation is linearlized according to a procedure applied by Beam and Warming and then, a tridiagonal system can be offered. Therefore, no iteration is necessary to solve the fluid dynamics equation. This leads great reduction of the CPU time. Also, POCHI1 can cope with non-linear boundary condition. In this study, comparison with experimental results was carried out. The normal zone propagation behavior was investigated in two samples of CIC conductors which had different hydraulic diameters. The measured and simulated normal zone propagation length showed relatively good agreement. However, the behavior of the normal voltage shows a little disagreement. These results indicate necessity to improve the treatment of the heat transfer coefficient in the turbulent flow region and the electric resistivity of the copper stabilizer in high temperature and high field region. (author)

155

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

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.

Bryson, Steve; Levy, Doron

2004-01-01

156

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Andreianov, Boris; Karlsen, Kenneth H; Pierre, Charles

2010-01-01

157

Three popular spatial differencing schemes for the discrete ordinates method are examined for two-dimensional Cartesian coordinates system. These are a positive, the step, and the diamond schemes. Contrary to the common belief that negative intensities will not occur when fine spatial discretization is used with the diamond scheme, under certain conditions, the diamond scheme will produce negative intensities irrespective of the number of control volums employed. The positive scheme can produce physically unrealistic trends. The diamond and positive schemes are also capable of producing physically unrealistic overshoots. In absorbing-emitting or absorbing-emitting-scattering media, grid refinement can result in negative intensities when the diamond or positive scheme is used.

Chai, John C.; Lee, Haeok S.; Patankar, Suhas V.

1993-01-01

158

Numerical time-domain electromagnetics based on finite-difference and convolution

Time-domain methods posses a number of advantages over their frequency-domain counterparts for the solution of wideband, nonlinear, and time varying electromagnetic scattering and radiation phenomenon. Time domain integral equation (TDIE)-based methods, which incorporate the beneficial properties of integral equation method, are thus well suited for solving broadband scattering problems for homogeneous scatterers. Widespread adoption of TDIE solvers has been retarded relative to other techniques by their inefficiency, inaccuracy and instability. Moreover, two-dimensional (2D) problems are especially problematic, because 2D Green's functions have infinite temporal support, exacerbating these difficulties. This thesis proposes a finite difference delay modeling (FDDM) scheme for the solution of the integral equations of 2D transient electromagnetic scattering problems. The method discretizes the integral equations temporally using first- and second-order finite differences to map Laplace-domain equations into the Z domain before transforming to the discrete time domain. The resulting procedure is unconditionally stable because of the nature of the Laplace- to Z-domain mapping. The first FDDM method developed in this thesis uses second-order Lagrange basis functions with Galerkin's method for spatial discretization. The second application of the FDDM method discretizes the space using a locally-corrected Nystrom method, which accelerates the precomputation phase and achieves high order accuracy. The Fast Fourier Transform (FFT) is applied to accelerate the marching-on-time process in both methods. While FDDM methods demonstrate impressive accuracy and stability in solving wideband scattering problems for homogeneous scatterers, they still have limitations in analyzing interactions between several inhomogenous scatterers. Therefore, this thesis devises a multi-region finite-difference time-domain (MR-FDTD) scheme based on domain-optimal Green's functions for solving sparsely-populated problems. The scheme uses a discrete Green's function (DGF) on the FDTD lattice to truncate the local subregions, and thus reduces reflection error on the local boundary. A continuous Green's function (CGF) is implemented to pass the influence of external fields into each FDTD region which mitigates the numerical dispersion and anisotropy of standard FDTD. Numerical results will illustrate the accuracy and stability of the proposed techniques.

Lin, Yuanqu

159

The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\\`ere equation. The approximation theory of Barles-Souganidis [Barles and Souganidis, Asymptotic Anal., 4 (1999) 271-283] requires that numerical schemes be monotone (or elliptic in the sense of [Oberman, SIAM J. Numer. Anal, 44 (2006) 879-895]. But such schemes have limited accuracy. In this article, we establish a convergence result for nearly monotone schemes. This allows us to construct finite difference discretizations of arbitrarily high-order. We demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth. In addition, the filtered scheme provides a natural detection principle for singularities. We employ this framework to construct a formally second-order scheme for the Monge-Amp\\`ere equation and present computational results on smooth and singular solutions.

Froese, Brittany D

2012-01-01

160

Directory of Open Access Journals (Sweden)

Full Text Available This article analyzes the changes in the number of cases of various clients of the pyramid and the establishment of the basic rules of the pyramid schemes based on discrete models. The article is also a continuation of previous work [1], which had formulas to simulate the amount collected by the pyramid scheme

Kovalenko A. V.

2012-10-01

161

Directory of Open Access Journals (Sweden)

Full Text Available Problem statement: Here, we develop a discretized scheme using only the penalty method without involving the multiplier parameter to examine the convergence and geometric ratio profiles. Approach: This approach reduces computational time arising from less data manipulation. Objectively, we wish to obtain a numerical solution comparing favourably with the analytic solution.. Methodologically, we discretize the given problem, obtain an unconstrained formulation and construct an operator which sets the stage for the application of the discretized extended conjugate gradient method. Results: We analyse the efficiency of the developed scheme by considering an example and examining the generated sequential approximate solutions and the convergence ratio profile computed quadratically per cycle using the discretized conjugate gradient method. Conclusion/Recommendations: Both results, as shown in the table, look comparably and this suggests that the developed scheme may very well approximate an analytic solution of a given problem to an appreciable level of tolerance without its prior knowledge.

O. Olotu

2011-01-01

162

Discrete unified gas kinetic scheme with force term for incompressible fluid flows

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...

Wu, Chen; Chai, Zhenhua; Wang, Peng

2014-01-01

163

A Review of High-Order and Optimized Finite-Difference Methods for Simulating Linear Wave Phenomena

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.

Zingg, David W.

1996-01-01

164

Compressed Semi-Discrete Central-Upwind Schemes for Hamilton-Jacobi Equations

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.

Bryson, Steve; Kurganov, Alexander; Levy, Doron; Petrova, Guergana

2003-01-01

165

An energy conserving finite-difference model of Maxwell's equations for soliton propagation

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.

Bachiri, H; Vázquez, L

1997-01-01

166

A free energy satisfying finite difference method for Poisson-Nernst-Planck equations

In this work we design and analyze a free energy satisfying finite difference method for solving Poisson-Nernst-Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.

Liu, Hailiang; Wang, Zhongming

2014-07-01

167

Original Signer's Forgery Attacks on Discrete Logarithm Based Proxy Signature Schemes

Directory of Open Access Journals (Sweden)

Full Text Available A proxy signature scheme enables a proxy signer to sign messages on behalf of the original signer. In this paper, we demonstrate that a number of discrete logarithm based proxy signature schemes are vulnerable to an original signer's forgery attack. In this attack, a malicious original signer can impersonate a proxy signer and produce a forged proxy signature on a message. A third party will incorrectly believe that the proxy signer was responsible for generating the proxy signature. This contradicts the strong unforgeability property that is required of proxy signatures schemes. We show six proxy signature schemes vulnerable to this attack including Lu et al.'s proxy blind multi-signature scheme, Xue and Cao's proxy blind signature scheme, Fu et al. and Gu et al.'s anonymous proxy signature schemes, Dai et al. and Huang et al.'s nominative proxy signature schemes are all insecure against the original signer's forgery.

Tianjie Cao

2007-05-01

168

To simulate seismic wave propagation in the spherical Earth, the Earth's curvature has to be taken into account. This can be done by solving the seismic wave equation in spherical coordinates by numerical methods. In this paper, we use an optimized, collocated-grid finite-difference scheme to solve the anisotropic velocity-stress equation in spherical coordinates. To increase the efficiency of the finite-difference algorithm, we use a non-uniform grid to discretize the computational domain. The grid varies continuously with smaller spacing in low velocity layers and thin layer regions and with larger spacing otherwise. We use stress-image setting to implement the free surface boundary condition on the stress components. To implement the free surface boundary condition on the velocity components, we use a compact scheme near the surface. If strong velocity gradient exists near the surface, a lower-order scheme is used to calculate velocity difference to stabilize the calculation. The computational domain is surrounded by complex-frequency shifted perfectly matched layers implemented through auxiliary differential equations (ADE CFS-PML) in a local Cartesian coordinate. We compare the simulation results with the results from the normal mode method in the isotropic and anisotropic models and verify the accuracy of the finite-difference method.

Zhang, Wei; Shen, Yang; Zhao, Li

2012-03-01

169

Numerical computation of transonic flows by finite-element and finite-difference methods

Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined.

Hafez, M. M.; Wellford, L. C.; Merkle, C. L.; Murman, E. M.

1978-01-01

170

The DFAs of Finitely Different Languages

Two languages are "finitely different" if their symmetric difference is finite. We consider the DFAs of finitely different regular languages and find major structural similarities. We proceed to consider the smallest DFAs that recognize a language finitely different from some given DFA. Such "f-minimal" DFAs are not unique, and this non-uniqueness is characterized. Finally, we offer a solution to the minimization problem of finding such f-minimal DFAs.

Badr, A; Badr, Andrew; Shipman, Ian

2007-01-01

171

For a Spacelab flight, a model experiment of the earth's atmospheric circulation has been proposed. This experiment is known as the Atmospheric General Circulation Experiment (AGCE). In the experiment concentric spheres will rotate as a solid body, while a dielectric fluid is confined in a portion of the gap between the spheres. A zero gravity environment will be required in the context of the simulation of the gravitational body force on the atmosphere. The present study is concerned with the development of pseudospectral/finite difference (PS/FD) model and its subsequent application to physical cases relevant to the AGCE. The model is based on a hybrid scheme involving a pseudospectral latitudinal formulation, and finite difference radial and time discretization. The advantages of the use of the hybrid PS/FD method compared to a pure second-order accurate finite difference (FD) method are discussed, taking into account the higher accuracy and efficiency of the PS/FD method.

Macaraeg, M. G.

1986-01-01

172

An Efficient Elliptic Curve Discrete Logarithm based Trapdoor Hash Scheme without Key Exposure

Directory of Open Access Journals (Sweden)

Full Text Available The trapdoor hash function plays essential role in constructing certain secure digital signature, and signature scheme that composed by trapdoor hash function is widely applied in different fields. However, the key exposure problem of trapdoor hash scheme has brought great distress. In this paper, an efficient trapdoor hash scheme without key exposure based on elliptic curve discrete logarithm is put forward and its security is analyzed, the scheme satisfies the five properties of trapdoor hash functions: effective calculation, trapdoor collision, collision resistance, key exposure resistance and semantic security. Through comparing and analyzing with the existing schemes, it shows that the proposed scheme, which has only multiplicative complexity and removes the operations of computing finite field element inverse, is more advantage in terms of safety and efficiency. Moreover, the scheme supports batch computation that it can greatly improve the efficiency of verification.

Yi Sun

2013-11-01

173

Energy Technology Data Exchange (ETDEWEB)

We present a comprehensive error estimation of four spatial discretization schemes of the two-dimensional Discrete Ordinates (SN) equations on Cartesian grids utilizing a Method of Manufactured Solution (MMS) benchmark suite based on variants of Larsen’s benchmark featuring different orders of smoothness of the underlying exact solution. The considered spatial discretization schemes include the arbitrarily high order transport methods of the nodal (AHOTN) and characteristic (AHOTC) types, the discontinuous Galerkin Finite Element method (DGFEM) and the recently proposed higher order diamond difference method (HODD) of spatial expansion orders 0 through 3. While AHOTN and AHOTC rely on approximate analytical solutions of the transport equation within a mesh cell, DGFEM and HODD utilize a polynomial expansion to mimick the angular flux profile across each mesh cell. Intuitively, due to the higher degree of analyticity, we expect AHOTN and AHOTC to feature superior accuracy compared with DGFEM and HODD, but at the price of potentially longer grind times and numerical instabilities. The latter disadvantages can result from the presence of exponential terms evaluated at the cell optical thickness that arise from the semianalytical solution process. This work quantifies the order of accuracy and the magnitude of the error of all four discretization methods for different optical thicknesses, scattering ratios and degrees of smoothness of the underlying exact solutions in order to verify or contradict the aforementioned intuitive expectation.

Sebastian Schunert; Yousry Y. Azmy; Damien Fournier

2011-05-01

174

A time-discretization scheme for coupled thermomechanical evolutions of shape memory alloys

In this Paper is introduced a time-discretization scheme for the numerical simulation of SMA structures. A particular feature of this time-discretized problem is that its solution can be expressed as solutions of a variational problem. This variational formulation allows one to study the existence and unicity of solutions. The approach presented is applied to simulate the propagation of a martensitic zone in a circular cylinder under traction.

Peigney, Michael

2006-03-01

175

WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes with a Lax-Wendroff time discretization procedure, termed WENO-LW schemes, for solving Hamilton-Jacobi equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. 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 the original WENO with Runge-Kutta time discretizations schemes (WENO-RK) of Jiang and Peng [G. Jiang, D. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126-2143] for Hamilton-Jacobi equations, the major advantages of WENO-LW schemes are more cost effective for certain problems and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

Qiu, Jianxian

2007-03-01

176

Finite strain plasticity: A discrete memory scheme and the associated numerical implementations

International Nuclear Information System (INIS)

The aim of this paper is twofold: first to present a covariant thermo-mechanical scheme of elasto-plastic hysteresis involving finite deformations. The definition of the scheme is obtained from a discrete memory Gibbs equation and by use of a dragged along material coordinate system; second to show that the use of differential geometry does not prevent the obtainment of numerical solutions for elasto-plastic problems involving finite deformations. (orig.)

177

International Nuclear Information System (INIS)

We develop here a new class of finite volume schemes on unstructured meshes for scalar conservation laws with stiff source terms. The schemes are of equilibrium type, hence with uniform bounds on approximate solutions, valid in cell entropy inequalities and exact for some equilibrium states. Convergence is investigated in the framework of kinetic schemes. Numerical tests show high computational efficiency and a significant advantage over standard cell centered discretization of source terms. Equilibrium type schemes produce accurate results even on test problems for which the standard approach fails. For some numerical tests they exhibit exponential type convergence rate. In two of our numerical tests an equilibrium type scheme with 441 nodes on a triangular mesh is more accurate than a standard scheme with 50002 grid points

178

A parallel adaptive finite difference algorithm for petroleum reservoir simulation

Energy Technology Data Exchange (ETDEWEB)

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)

Hoang, Hai Minh

2005-07-01

179

High Order Finite Difference Methods for Multiscale Complex Compressible Flows

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.

Sjoegreen, Bjoern; Yee, H. C.

2002-01-01

180

Finite difference methods for the solution of unsteady potential flows

A brief review is presented of various problems which are confronted in the development of an unsteady finite difference potential code. This review is conducted mainly in the context of what is done for a typical small disturbance and full potential methods. The issues discussed include choice of equation, linearization and conservation, differencing schemes, and algorithm development. A number of applications including unsteady three-dimensional rotor calculation, are demonstrated.

Caradonna, F. X.

1985-01-01

181

A unified formalism for spatial discretization schemes of transport equations in slab geometry

International Nuclear Information System (INIS)

It is shown that most of the spatial discretization schemes of transport equations in slab geometry which have been developed recently are particular applications of a general finite element oriented formalism developed by this author and his collaborators for the numerical integration of systems of stiff ordinary differential equations. (author)

182

Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation

Directory of Open Access Journals (Sweden)

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.

Walter H. Aschbacher

2009-01-01

183

An Efficient Signcryption Scheme based on The Elliptic Curve Discrete Logarithm Problem

Directory of Open Access Journals (Sweden)

Full Text Available Elliptic Curve Cryptosystems (ECC have recently received significant attention by researchers due to their performance. Here, an efficient signcryption scheme based on elliptic curve will be proposed, which can effectively combine the functionalities of digital signature and encryption. Since the security of the proposed method is based on the difficulty of solving discrete logarithm over an elliptic curve. The purposes of this paper are to demonstrate how to specify signcryption scheme on elliptic curves over finite field, and to examine the efficiency of such scheme. The results analysis are explained.

Fatima Amounas

2013-02-01

184

Second-order accurate nonoscillatory schemes for scalar conservation laws

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.

Huynh, Hung T.

1989-01-01

185

Finite-difference methods for eigenvalues

International Nuclear Information System (INIS)

Some useful ways of improving the speed and accuracy of finite-difference methods for eigenvalue calculations are proposed and are tested successfully on several problems, including one for which the potential is highly singular at the origin. (author)

186

A singular finite difference treatment of re-entrant corner flow. Pt. 1

International Nuclear Information System (INIS)

Moffatt has given a series solution for the steady flow of a viscous fluid in the neighbourhood of a sharp corner. A method of incorporating this solution into a finite difference scheme is described here, for the case of a rectangular re-entrant corner. Only minor adjustments to the usual finite difference scheme are required. Computational results for the flow through a contraction geometry are given. (orig.)

187

International Nuclear Information System (INIS)

We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. In particular, we show that the corresponding restrictions on the finite-gap data are of the same type as those for the fixed energy problem for the two-dimensional Schroedinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R3

188

A Modified Third-Order Semi-Discrete Central-Upwind Scheme for MHD Simulation

The Kurganov scheme is a third-order semi-discrete central numerical algorithm. The high solution of the scheme is ensured by a piecewise quadratic non-oscillatory reconstruction which consists of the cell-average data. We employ a modification of the smooth limiter of reconstruction in a simple way. The modified limiter possesses rigorous positivity and the reformulation does not change the non-oscillatory property of reconstruction. In order to explore the potential capability of application of the modified Kurganov scheme to magnetohydrodynamics (MHD) and resistive magnetohydrodynamics (RMHD) equations, two numerical problems are simulated in two dimensions (2D). These numerical simulations demonstrate that the modified Kurganov scheme keeps high precision and has stable reliable results for MHD and RMHD applications.

Ji, Zhen; Zhou, Yu-Fen; Hou, Tian-Xiang

2011-07-01

189

A Modified Third-Order Semi-Discrete Central-Upwind Scheme for MHD Simulation

International Nuclear Information System (INIS)

The Kurganov scheme is a third-order semi-discrete central numerical algorithm. The high solution of the scheme is ensured by a piecewise quadratic non-oscillatory reconstruction which consists of the cell-average data. We employ a modification of the smooth limiter of reconstruction in a simple way. The modified limiter possesses rigorous positivity and the reformulation does not change the non-oscillatory property of reconstruction. In order to explore the potential capability of application of the modified Kurganov scheme to magnetohydrodynamics (MHD) and resistive magnetohydrodynamics (RMHD) equations, two numerical problems are simulated in two dimensions (2D). These numerical simulations demonstrate that the modified Kurganov scheme keeps high precision and has stable reliable results for MHD and RMHD applications. (physics of gases, plasmas, and electric discharges)

190

Convergence of Finite Difference Methods for Poisson's Equation with Interfaces

In this paper, a weak formulation of the discontinuous variable coefficient Poisson equation with interfacial jumps is studied. The existence, uniqueness and regularity of solutions of this problem are obtained. It is shown that the application of the Ghost Fluid Method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the Ghost Fluid Method is proven to be convergent.

Liu, X D; Liu, Xu-Dong; Sideris, Thomas C.

2001-01-01

191

Flow problems, finite differences and the frontal method

The frontal solution method has been used for the solution of large sparse finite element systems. These systems are assemblages of element stiffness matrices. This property is exploited by the frontal method which alternates between generation of element stiffness matrices and the elimination of variables from the system. The reported investigation is concerned with the application of the frontal method to the nonsymmetric system of equations resulting from a finite difference discretization of a five equation transient fluids model. Attention is given to the fluids model and implicit difference system, a dual variable method, and the frontal solution of the dual variable system.

Amit, R.; Cullen, C.; Hall, C.; Mesina, G.; Porsching, T.

192

Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation

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.

Koley, U

2011-01-01

193

Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models

A unifying scheme based on an ancestor model is proposed for generating a wide rage of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sine-Gordon, NLS, derivative NLS, Liouville model, (non-)relativistic Toda chain etc. Our scheme introduces a novel possibility of building integrable hybrid systems including multi-component models by combining different descendant models. We also generate new inhomogeneous models like variable mass sine-Gordon, variable coefficient NLS, Ablowitz-Ladik model, Toda chains etc. keeping their flows isospectral, as opposed to the standard approach. All our models are generated from the same ancestor model (or its $q \\to 1$ limit) and satisfy the classical Yang-Baxter equation sharing the same $r$-matrix. This reveals a deep universality in these diverse systems, which become explicit at their action-angle level.

Kundu, A

2002-01-01

194

Due to limitations of computers, prediction of structure-borne sound remains difficult for large-scale problems. Herein a prediction method for low-frequency structure-borne sound transmissions on concrete structures using the finite-difference time-domain scheme is proposed. The target structure is modeled as a composition of multiple plate elements to reduce the dimensions of the simulated vibration field from three-dimensional discretization by solid elements to two-dimensional discretization. This scheme reduces both the calculation time and the amount of required memory. To validate the proposed method, the vibration characteristics using the numerical results of the proposed scheme are compared to those measured for a two-level concrete structure. Comparison of the measured and simulated results suggests that the proposed method can be used to simulate real-scale structures. PMID:25190384

Asakura, T; Ishizuka, T; Miyajima, T; Toyoda, M; Sakamoto, S

2014-09-01

195

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Directory of Open Access Journals (Sweden)

Full Text Available The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.

J. M. Guevara-Jordan

2007-06-01

196

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Directory of Open Access Journals (Sweden)

Full Text Available The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.

Rojas S

2007-01-01

197

Finite Difference Migration Imaging of Magnetotellurics

Directory of Open Access Journals (Sweden)

Full Text Available we put forward a new migration imaging technique of Magnetotellurics (MT data based on improved finite difference method, which increased the accuracy of difference equation and imaging resolution greatly. We also discussed the determination of background resistivity and reimaging. The processing results of theoretical model and case study indicated that this method was a more practical and effective for MT imaging. Finally the characteristics of finite difference migration imaging were summarized and the factors which can affect the migration imaging were analyzed.

Runlin Du

2013-10-01

198

Spatial Discretization of the Shallow Water Equations in Spherical Geometry Using Osher's Scheme

The shallow water equations in spherical geometry provide a first prototype for developing and testing numerical algorithms for atmospheric circulation models. Since the seventies these models have often been solved with spectral methods. Increasing demands on grid resolution combined with massive parallelism and local grid refinement seem to offer significantly better perspectives for gridpoint methods. In this paper we study the use of Osher's finite-volume scheme for the spatial discretization of the shallow water equations on the rotating sphere. This finite volume scheme of upwind type is well suited for solving a hyperbolic system of equations. Special attention is paid to the pole problem. To that end Osher's scheme is applied on the common (reduced) latitude-longitude grid and on a stereographic grid. The latter is most appropriate in the polar region as in stereographic coordinates the pole singularity does not exist. The latitude-longitude grid is preferred on lower latitudes. Therefore, across the sphere we apply Osher's scheme on a combined grid connecting the two grids at high latitude. We will show that this provides an attractive spatial discretization for explicit integration methods, as it can greatly reduce the time step limitation incurred by the pole singularity when using a latitude-longitude grid only. When time step limitation plays no significant role, the standard (reduced) latitude-longitude grid is advocated provided that the grid is kept sufficiently fine in the polar region to resolve flow over the poles.

Lanser, D.; Blom, J. G.; Verwer, J. G.

2000-12-01

199

A finite-difference lattice Boltzmann approach for gas microflows

Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.

Ghiroldi, G P

2013-01-01

200

In this paper, we explore the Lax-Wendroff (LW) type time discretization as an alternative procedure to the high order Runge-Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in [3,5]. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy-Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes [3] and third-order spatial discretization using curvilinear meshes [5]. Comparing with the Runge-Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.

Liu, Wei; Cheng, Juan; Shu, Chi-Wang

2009-12-01

201

Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates

Digital Repository Infrastructure Vision for European Research (DRIVER)

We construct a new finite difference method for the flow of ideal viscous isentropic gas in one spatial dimension. For the continuity equation, the method is a standard upwind discretization. For the momentum equation, the method is an uncommon upwind discretization, where the moment and the velocity are solved on dual grids. Our main result is convergence of the method as discretization parameters go to zero. Convergence is proved by adapting the mathematical existence theo...

Karper, Trygve K.

2013-01-01

202

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting...

Young, P; Martinsson, P G

2012-01-01

203

On the wavelet optimized finite difference method

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.

Jameson, Leland

1994-01-01

204

The numerical solution of problems in calculus of variation using Chebyshev finite difference method

International Nuclear Information System (INIS)

The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results

205

A Two-Level Classification Scheme for CDHMM-Based Discrete-Utterance Recognition

Directory of Open Access Journals (Sweden)

Full Text Available In the paper a method of speeding up the response of a CDHMM based speech recognition system is introduced. The method, applicable for the recognition of discrete utterances, uses a two-level classification scheme. It consists in a fast match done with simplified models, followed by a final accurate match with a limited number of selected standard models. In this way the recognition time can be reduced by great deal without any significant loss of recognition accuracy. The method has been successfully applied in the design of real-time speech recognition systems operating with small and middle-size vocabularies.

J. Nouza

1996-04-01

206

Benchmark 3D: CeVe-DDFV, a discrete duality scheme with cell/vertex unknowns.

Digital Repository Infrastructure Vision for European Research (DRIVER)

The "3D anisotropy Benchmark" (http://www.latp.univ-mrs.fr/latp_numerique/?q=node/4) addresses a three-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or discontinuous. This paper presents numerical results to this benchmark for a 3D Cell and Vertex centered DDFV scheme. The method applies to very general 3D meshes including non conformal o...

Coudiere, Yves; Pierre, Charles

2011-01-01

207

Channel Estimation Scheme with Low-Complexity Discrete Cosine Transform in MIMO-OFDM System

Channel estimation is a key baseband processing task in wireless systems. Filtering or smoothing algorithms can improve the accuracy of channel estimates and the Discrete Cosine Transform (DCT) can be used for this purpose. By using the DCT, performance will be improved compared to the straight-forward approach of per subcarrier estimation (PSE). However, the complexity of the DCT is not negligible. This paper proposes a low-complexity channel estimation scheme using the DCT. Simulation results show that the performance is improved by more than 1dB compared with PSE in MIMO-OFDM system.

Takeda, Daisuke; Tanabe, Yasuhiko

208

High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equations

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-Noelle-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spatial 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.

Bryson, Steve; Levy, Doron

2003-07-01

209

International Nuclear Information System (INIS)

In this article, a number of due changes in both a discrete ordinates neutron transport model and a companion void fraction evaluation scheme are introduced. These changes are due to the explicit consideration of a weakly divergent neutron beam, i.e. a neutron beam consisting of a monodirectional component with normal incidence, which is modeled by a Dirac delta distribution, and an angularly continuous component, which is modeled by a smooth function of the angular variable. Computational tests are performed in order to illustrate the numerical consistency of the evaluation scheme for weakly divergent beams with respect to the order of angular quadrature, as well as its low sensitivity to experimental inaccuracies in the detector responses. (orig.)

210

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.

Swisshelm, Julie M.

1989-01-01

211

Finite difference schemes for second order systems describing black holes

In the harmonic description of general relativity, the principle part of Einstein's equations reduces to 10 curved space wave equations for the componenets of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.

Motamed, M; Szilágyi, B; Kreiss, H O; Winicour, J; Motamed, Mohammad

2006-01-01

212

An induced charge readout scheme incorporating image charge splitting on discrete pixels

International Nuclear Information System (INIS)

Top hat electrostatic analysers used in space plasma instruments typically use microchannel plates (MCPs) followed by discrete pixel anode readout for the angular definition of the incoming particles. Better angular definition requires more pixels/readout electronics channels but with stringent mass and power budgets common in space applications, the number of channels is restricted. We describe here a technique that improves the angular definition using induced charge and an interleaved anode pattern. The technique adopts the readout philosophy used on the CRRES and CLUSTER I instruments but has the advantages of the induced charge scheme and significantly reduced capacitance. Charge from the MCP collected by an anode pixel is inductively split onto discrete pixels whose geometry can be tailored to suit the scientific requirements of the instrument. For our application, the charge is induced over two pixels. One of them is used for a coarse angular definition but is read out by a single channel of electronics, allowing a higher rate handling. The other provides a finer angular definition but is interleaved and hence carries the expense of lower rate handling. Using the technique and adding four channels of electronics, a four-fold increase in the angular resolution is obtained. Details of the scheme and performance results are presented

213

Optimal Independent Encoding Schemes for Several Classes of Discrete Degraded Broadcast Channels

Let $X \\to Y \\to Z$ be a discrete memoryless degraded broadcast channel (DBC) with marginal transition probability matrices $T_{YX}$ and $T_{ZX}$. For any given input distribution $\\boldsymbol{q}$, and $H(Y|X) \\leq s \\leq H(Y)$, define the function $F^*_{T_{YX},T_{ZX}}(\\boldsymbol{q},s)$ as the infimum of $H(Z|U)$ with respect to all discrete random variables $U$ such that a) $H(Y|U) = s$, and b) $U$ and $Y,Z$ are conditionally independent given $X$. This paper studies the function $F^*$, its properties and its calculation. This paper then applies these results to several classes of DBCs including the broadcast Z channel, the input-symmetric DBC, which includes the degraded broadcast group-addition channel, and the discrete degraded multiplication channel. This paper provides independent encoding schemes and demonstrates that each achieve the boundary of the capacity region for the corresponding class of DBCs. This paper first represents the capacity region of the DBC $X \\to Y \\to Z$ with the function $F^*_{T...

Xie, Bike

2008-01-01

214

Finite difference methods for coupled flow interaction transport models

Directory of Open Access Journals (Sweden)

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.

Shelly McGee

2009-04-01

215

In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

Hejranfar, Kazem; Ezzatneshan, Eslam

2014-06-01

216

Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation

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,…

Prentice, J. S. C.

2012-01-01

217

Discrete unified gas kinetic scheme for all Knudsen number flows: II. Compressible case

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...

Guo, Zhaoli; Xu, Kun

2014-01-01

218

Robust discretizations versus increase of the time step for the Lorenz system.

When continuous systems are discretized, their solutions depend on the time step chosen a priori. Such solutions are not necessarily spurious in the sense that they can still correspond to a solution of the differential equations but with a displacement in the parameter space. Consequently, it is of great interest to obtain discrete equations which are robust even when the discretization time step is large. In this paper, different discretizations of the Lorenz system are discussed versus the values of the discretization time step. It is shown that the sets of difference equations proposed are more robust versus increases of the time step than conventional discretizations built with standard schemes such as the forward Euler, backward Euler, or centered finite difference schemes. The nonstandard schemes used here are Mickens' scheme and Monaco and Normand-Cyrot's scheme. PMID:15836264

Letellier, Christophe; Mendes, Eduardo M A M

2005-03-01

219

Finite difference methods for anomalous diffusion

Fractional diffusion equations extend the classical models by substituting fractional derivatives in place of the usual integer order derivatives. Fractional equations have proven useful in modeling tracers in ground water and surface water, and are finding new applications in diverse areas including geomorphology, heat transfer, sound conduction in human tissue, and finance. Available solution methods include semi-analytical transform inversion, particle tracking, finite element, and finite difference methods. This talk will describe stable and consistent implicit Euler methods in sufficient detail for coding. As in the classical case, correctly formulated implicit Euler methods are unconditionally stable, while explicit Euler methods are stable under a significant step size restriction. Tested FORTRAN codes are freely available, and example runs will be presented, in which numerical solutions are compared to exact analytical solutions. Model extensions and open problems will be outlined.

Meerschaert, M. M.; Tadjeran, C.

2008-12-01

220

Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling

Directory of Open Access Journals (Sweden)

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.

Matti Karjalainen

2004-06-01

221

Spectral-finite difference algorithm for three dimensional incompressible MHD

International Nuclear Information System (INIS)

A code was developed that solves the three dimensional incompressible magnetohydrodynamics equations in a cylindrical geometry. The algorithm differs from those used in some of the existing 3-D codes in that no assumption is made about ordering of various quantities; thus, it is applicable for arbitrary beta and aspect ratios. However, we have retained the incompressibility condition in order to eliminate the fast Alfven time scale. Variables are expanded in Fourier series in the azimuthal and axial coordinates, which reduces the three dimensional MHD equations to one dimensional equations in the radial coordinate for each of the Fourier harmonics. These resulting equations are integrated using a finite-difference scheme in the radial direction

222

Finite Difference Method of the Study on Radioactivities DispersionModeling in Environment of Ground

International Nuclear Information System (INIS)

It has been resulted the mathematics equation as model of constructingthe computer algorithm deriving from the transport equation having been theform of radionuclides dispersion in the environment of ground as a result ofdiffusion and advection process. The derivation of mathematics equation usedthe finite difference method into three schemes, the explicit scheme,implicit scheme and Crank-Nicholson scheme. The computer algorithm then wouldbe used as the basic of making the software in case of making a monitoringsystem of automatic radionuclides dispersion on the area around the nuclearfacilities. By having the three schemes, so it would be, in choosing thesoftware system, able to choose the more approximate with the fact. (author)

223

A novel variable-grid finite-difference method for fiber lasers

A novel finite-difference method with a variable discrete grid is proposed. The algorithm is applied to the traveling wave model for fiber lasers successfully. The simulation for the CW and the actively Q-switched output of Yb-doped fiber laser is compared with the reported results and the results simulated by a traditional constant-grid finite difference method. The comparisons prove that the new algorithm causes negligible adverse impacts, while the efficiency is largely improved. The novel algorithm presented in this paper can be optimized to be a promising method for solving other sorts of partial differential equations.

Liu, Zhen

2014-11-01

224

Perfectly Matched Layer for the Wave Equation Finite Difference Time Domain Method

The perfectly matched layer (PML) is introduced into the wave equation finite difference time domain (WE-FDTD) method. The WE-FDTD method is a finite difference method in which the wave equation is directly discretized on the basis of the central differences. The required memory of the WE-FDTD method is less than that of the standard FDTD method because no particle velocity is stored in the memory. In this study, the WE-FDTD method is first combined with the standard FDTD method. Then, Berenger's PML is combined with the WE-FDTD method. Some numerical demonstrations are given for the two- and three-dimensional sound fields.

Miyazaki, Yutaka; Tsuchiya, Takao

2012-07-01

225

TWO STAGE DISCRETE TIME EXTENDED KALMAN FILTER SCHEME FOR MICRO AIR VEHICLE

Directory of Open Access Journals (Sweden)

Full Text Available Navigation of Micro Air Vehicle (MAV is one of the most challenging areas of twenty first century’s research. Micro Air Vehicle (MAV is the miniaturized configuration of aircraft with a size of six inches in length and below the weight of hundred grams, which includes twenty grams of payload as well. Due to its small size, MAV is highly affected by the wind gust and therefore the navigation of Micro Air Vehicle (MAV is very important because precise navigation is a very basic step for the control of the Micro Air Vehicle (MAV. This paper presents two stage cascaded discrete time Extended Kalman Filter while using INS/GPS based navigation. First stage of this scheme estimates the Euler angles of Micro Air Vehicle (MAV whereas the second stage of this scheme estimates the position of Micro Air Vehicle (MAV in terms of height, longitude and latitude. As the system is considered as non-linear, so Extended Kalman Filter is used. On-board sensors in first stage included MEMS Gyro, MEMS Accelerometer, MEMS Magnetometer whereas second stage includes GPS.

Ali Usman

2012-03-01

226

High Order Finite Difference Methods, Multidimensional Linear Problems and Curvilinear Coordinates

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.

Nordstrom, Jan; Carpenter, Mark H.

1999-01-01

227

The reliability of finite difference and particle methods for fragmentation problems

International Nuclear Information System (INIS)

The authors reply to criticisms made by Boss and Bodenheimer of the use of their particle method for fragmentation problems. A new calculation with their particle scheme is discussed, which indicates that, in the event of drastic angular momentum loss, the particle method results resemble more closely those from the finite difference calculations. (U.K.)

228

International Nuclear Information System (INIS)

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

229

Stability of pseudospectral and finite-difference methods for variable coefficient problems

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.

Gottlieb, D.; Orszag, S. A.; Turkel, E.

1981-01-01

230

This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.

Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.

1983-01-01

231

Stochastic finite-difference time-domain

This dissertation presents the derivation of an approximate method to determine the mean and the variance of electro-magnetic fields in the body using the Finite-Difference Time-Domain (FDTD) method. Unlike Monte Carlo analysis, which requires repeated FDTD simulations, this method directly computes the variance of the fields at every point in space at every sample of time in the simulation. This Stochastic FDTD simulation (S-FDTD) has at its root a new wave called the Variance wave, which is computed in the time domain along with the mean properties of the model space in the FDTD simulation. The Variance wave depends on the electro-magnetic fields, the reflections and transmission though the different dielectrics, and the variances of the electrical properties of the surrounding materials. Like the electro-magnetic fields, the Variance wave begins at zero (there is no variance before the source is turned on) and is computed in the time domain until all fields reach steady state. This process is performed in a fraction of the time of a Monte Carlo simulation and yields the first two statistical parameters (mean and variance). The mean of the field is computed using the traditional FDTD equations. Variance is computed by approximating the correlation coefficients between the constituitive properties and the use of the S-FDTD equations. The impetus for this work was the simulation time it takes to perform 3D Specific Absorption Rate (SAR) FDTD analysis of the human head model for cell phone power absorption in the human head due to the proximity of a cell phone being used. In many instances, Monte Carlo analysis is not performed due to the lengthy simulation times required. With the development of S-FDTD, these statistical analyses could be performed providing valuable statistical information with this information being provided in a small fraction of the time it would take to perform a Monte Carlo analysis.

Smith, Steven Michael

232

Energy Technology Data Exchange (ETDEWEB)

The coarse mesh finite difference (CMFD) formulation has been applied to Monte Carlo (MC) simulations in order to mitigate the issue of large real variances of pin power tallies in full-core problems. In this work, a parallelized multigroup (MG) two-dimensional (2-D) MC code named PRIDE (Probabilistic Reactor Investigation with Discretized Energy), which is capable of handling lattices of square pin cells within which circular substructures can be modeled, has been developed as a tool for the investigations of the new method. In this code, a scheme to construct a CMFD linear system is based on the MC tallies of coarse mesh average fluxes and the net currents at coarse mesh interfaces. These tallies are accumulated over the MC cycles to get more stable CMFD solutions which are used for feedback to MC fission source distribution (FSD). The feedback scheme in this code employs a weight adjustment of fission source neutrons for the next MC cycle that is to reflect the global CMFD FSD into the MC FSD. The performance of CMFD feedback has been investigated in terms of the number of inactive cycles required for the convergence of FSD and also the reduction of real variances of local property tallies in active cycles. The applications to 2-D multigroup full-core pressurized water reactor problems have demonstrated that the MC FSD converges considerably faster and the real variances of pin powers are smaller by a factor of 4 with CMFD FSD feedback. It is also noted that the large real variances of pin powers are caused mainly by the global assembly-wise fluctuations of power distributions in a large core rather than local fluctuations.

Lee, Min-Jae [ORNL; Joo, Han Gyu [Seoul National University; Lee, Deokjung [ORNL; Smith, Kord [Studsvik Scandpower, Inc.

2010-01-01

233

International Nuclear Information System (INIS)

The coarse mesh finite difference (CMFD) formulation has been applied to Monte Carlo (MC) simulations in order to mitigate the issue of large real variances of pin power tallies in full-core problems. In this work, a parallelized multigroup (MG) two-dimensional (2-D) MC code named PRIDE (Probabilistic Reactor Investigation with Discretized Energy), which is capable of handling lattices of square pin cells within which circular substructures can be modeled, has been developed as a tool for the investigations of the new method. In this code, a scheme to construct a CMFD linear system is based on the MC tallies of coarse mesh average fluxes and the net currents at coarse mesh interfaces. These tallies are accumulated over the MC cycles to get more stable CMFD solutions which are used for feedback to MC fission source distribution (FSD). The feedback scheme in this code employs a weight adjustment of fission source neutrons for the next MC cycle that is to reflect the global CMFD FSD into the MC FSD. The performance of CMFD feedback has been investigated in terms of the number of inactive cycles required for the convergence of FSD and also the reduction of real variances of local property tallies in active cycles. The applications to 2-D multigroup full-core pressurized water reactor problems have demonstrated that the MC FSD converges considerably faster and the real variances of pin powers are smaller by a factor of 4 with CMFD FSD feedback. It is also noted that the laFSD feedback. It is also noted that the large real variances of pin powers are caused mainly by the global assembly-wise fluctuations of power distributions in a large core rather than local fluctuations.

234

International Nuclear Information System (INIS)

The coarse mesh finite difference (CMFD) formulation has been applied to Monte Carlo (MC) simulations in order to mitigate the issue of large real variances of pin power tallies in full-core problems. In this work, a parallelized multigroup (MG) two-dimensional (2-D) MC code named PRIDE (Probabilistic Reactor Investigation with Discretized Energy), which is capable of handling lattices of square pin cells within which circular substructures can be modeled, has been developed as a tool for the investigations of the new method. In this code, a scheme to construct a CMFD linear system is based on the MC tallies of coarse mesh average fluxes and the net currents at coarse mesh interfaces. These tallies are accumulated over the MC cycles to get more stable CMFD solutions which are used for feedback to MC fission source distribution (FSD). The feedback scheme in this code employs a weight adjustment of fission source neutrons for the next MC cycle that is to reflect the global CMFD FSD into the MC FSD. The performance of CMFD feedback has been investigated in terms of the number of inactive cycles required for the convergence of FSD and also the reduction of real variances of local property tallies in active cycles. The applications to 2-D multigroup full-core pressurized water reactor problems have demonstrated that the MC FSD converges considerably faster and the real variances of pin powers are smaller by a factor of 4 with CMFD FSD feedback. It is also noted that the laFSD feedback. It is also noted that the large real variances of pin powers are caused mainly by the global assembly-wise fluctuations of power distributions in a large core rather than local fluctuations. (author)

235

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes [5,25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.

Qiu, Jing-Mei; Shu, Chi-Wang

2011-02-01

236

Skew-symmetric convection form and secondary conservative finite difference methods for moving grids

The secondary conservative finite difference method for the convective term is recognized as a useful tool for unsteady flow simulations. However, the secondary conservative convection scheme and associated skew-symmetric form have not been extended to those for moving grids. In this study, the skew-symmetric form and the secondary conservative convection schemes for ALE type moving grid simulations are proposed. For the moving grid simulations, the geometric conservation law (GCL) for metrics and the Jacobian is known as a mathematical constraint for capturing a uniform flow. A new role of the GCL is revealed in association with the commutability and conservation properties of the convection schemes. The secondary conservative convection schemes for moving grids are then constructed for compressible and incompressible flows, respectively. For compressible flows, it is necessary to introduce a shock capturing method to resolve discontinuities. However, the shock capturing methods do not work well for turbulent flow simulations because of their excessive numerical dissipation. On the other hand, the secondary conservative finite difference method does not work well for flows with discontinuities. In this study, we also present a computational technique that combines the shock capturing and the secondary conservative finite difference methods. In order to check the commutability and conservation properties of the convection schemes, numerical tests are done for compressible and incompressible inviscid periodic flows on moving grids. Then, the reliabilities of the schemes are demonstrated on the piston problem, the flow around pitching airfoil, and the flow around an oscillating square cylinder.

Morinishi, Yohei; Koga, Kazuki

2014-01-01

237

Test of two methods for faulting on finite-difference calculations

Tests of two fault boundary conditions show that each converges with second order accuracy as the finite-difference grid is refined. The first method uses split nodes so that there are disjoint grids that interact via surface traction. The 3D version described here is a generalization of a method I have used extensively in 2D; it is as accurate as the 2D version. The second method represents fault slip as inelastic strain in a fault zone. Offset of stress from its elastic value is seismic moment density. Implementation of this method is quite simple in a finite-difference scheme using velocity and stress as dependent variables.

Andrews, D.J.

1999-01-01

238

Approximate Lie Group Analysis of Finite-difference Equations

Digital Repository Infrastructure Vision for European Research (DRIVER)

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 algebr...

Latypov, Azat M.

1995-01-01

239

Using finite difference method to simulate casting thermal stress

Directory of Open Access Journals (Sweden)

Full Text Available Thermal stress simulation can provide a scientific reference to eliminate defects such as crack, residual stress centralization and deformation etc., caused by thermal stress during casting solidification. To study the thermal stress distribution during casting process, a unilateral thermal-stress coupling model was employed to simulate 3D casting stress using Finite Difference Method (FDM, namely all the traditional thermal-elastic-plastic equations are numerically and differentially discrete. A FDM/FDM numerical simulation system was developed to analyze temperature and stress fields during casting solidification process. Two practical verifications were carried out, and the results from simulation basically coincided with practical cases. The results indicated that the FDM/FDM stress simulation system can be used to simulate the formation of residual stress, and to predict the occurrence of hot tearing. Because heat transfer and stress analysis are all based on FDM, they can use the same FD model, which can avoid the matching process between different models, and hence reduce temperature-load transferring errors. This approach makes the simulation of fluid flow, heat transfer and stress analysis unify into one single model.

Liao Dunming

2011-05-01

240

Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\\'ere equation

The elliptic Monge-Amp\\`ere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Amp\\'ere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the nece...

Froese, Brittany D

2010-01-01

241

Improvement of the finite difference lattice Boltzmann method for low mach number flows

International Nuclear Information System (INIS)

This paper presents a numerical method to compute flow-acoustic resonance at low Mach number within a reasonable computing time. Light water reactors have experienced flow-acoustic resonance which is attributed to unsteady compressible flows at low Mach number. This phenomenon is undesirable because the induced sound causes loud noise and vibrations of the mechanical structures. However, a numerical simulation of this flow-acoustic resonance at low Mach number requires a large computing time and a highly accurate method in order to simultaneously compute flows and acoustic waves. The finite difference lattice Boltzmann method which is a powerful tool for obtaining computational fluid dynamics has high accuracy for simultaneous calculation of flows and acoustic waves. It becomes an efficient method to compute low Mach number flow if the computing time is shortened. In this paper, the finite difference lattice Boltzmann method was sped up. Three improvements were proposed: development of a new particle model, modification of the governing equation, and employment of an efficient time marching scheme. The computing time of the proposed finite difference lattice Boltzmann model was compared with the conventional finite difference lattice Boltzmann model for the calculation of the cubic cavity flow. The results showed that the computing time of the proposed model is 30% of the time needed by the conventional finite difference lattice Boltzmann model. The flow-acoustic resonance at low Mach number at the side branch was calculated using the proposed model. The numerical results showed quantitative agreement with the experimental data. (author)

242

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.

von Sydow, Lina

2013-10-01

243

On the preservation of phase space structure under multisymplectic discretization

International Nuclear Information System (INIS)

In this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and Fourier spectral approximations. Multisymplectic (MS) schemes are developed for two benchmark nonlinear wave equations, the sine-Gordon and nonlinear Schroedinger equations. We examine the implications of preserving the MS structure under discretization on the numerical scheme's ability to preserve phase space structure, as measured by the nonlinear spectrum of the governing equation. We find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws, dynamical invariants and complicated phase space structures

244

Discrete Filters for Large Eddy Simulation of Forced Compressible MHD Turbulence

Digital Repository Infrastructure Vision for European Research (DRIVER)

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 ...

Chernyshov, Alexander A.; Karelsky, Kirill V.; Petrosyan, Arakel S.

2013-01-01

245

In this paper, two explicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation with boundary values which are functions. The fractional derivative is treated by applying shifted Grünwald-Letnikov formula of order ? ?(0,1) while the first and second order spatial derivatives are replaced by the corresponding finite difference approximations. The stability analysis is investigated via von Neumann method. Numerical results are presented to demonstrate the effectiveness of the schemes.

Al-Shibani, F. S.; Ismail, A. I. Md.; Abdullah, F. A.

2014-07-01

246

Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.

Mccoy, M. J.

1980-01-01

247

Directory of Open Access Journals (Sweden)

Full Text Available Most of the localization algorithms in past decade are usually based on Monte Carlo, sequential monte carlo and adaptive monte carlo localization method. In this paper we proposed a new scheme called DQMCL which employs the antithetic variance reduction method to improve the localization accuracy. Most existing SMC and AMC based localization algorithm cannot be used in dynamic sensor network but DQMCL can work well even without need of static sensor network with the help of discrete power control method for the entire sensor to improve the average Localization accuracy. Also we analyse a quasi monte carlo method for simulating a discrete time antithetic markov time steps to improve the life time of the sensor node. Our simulation result shows that overall localization accuracy will be more than 88% and localization error is below 35% with synchronization error observed at different discrete time interval.

M.Vasim babu

2014-05-01

248

Digital Repository Infrastructure Vision for European Research (DRIVER)

Nonlinear systems, such as switching DC-DC boost or buck converters, have rich dynamics. A simple one-dimensional discrete-time model is used to analyze the boost or buck converter in discontinuous conduction mode. Seven different control schemes (open-loop power stage, voltage mode control, current mode control, constant power load, constant current load, constant-on-time control, and boundary conduction mode) are analyzed systematically. The linearized dynamics is obtained...

Fang, Chung-chieh

2012-01-01

249

Efficient architectures for two-dimensional discrete wavelet transform using lifting scheme.

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

Xiong, Chengyi; Tian, Jinwen; Liu, Jian

2007-03-01

250

The fourth-order finite-difference scheme with fully implicit time-marching presently used to computationally study the spatial instability of planar Poiseuille flow incorporates a novel treatment for outflow boundary conditions that renders the buffer area as short as one wavelength. A semicoarsening multigrid method accelerates convergence for the implicit scheme at each time step; a line-distributive relaxation is developed as a robust fast solver that is efficient for anisotropic grids. Computational cost is no greater than that of explicit schemes, and excellent agreement with linear theory is obtained.

Liu, C.; Liu, Z.

1993-01-01

251

Equivalent constants to be used in finite difference diffusion calculations

International Nuclear Information System (INIS)

Nowadays, we use very accurate transport codes for the fine treatment, in space and energy, of the very heterogeneous reactor lattices. However, such a procedure is prohibitively expensive for the core calculations and we only use it for the calculation of small subregions of the reactor (assemblies). The whole core calculation is generally performed using few groups diffusion codes on a rather simplified geometry. This calculation scheme implies the previous determination of broad-groups constants for homogenized regions. One of the most common methods to do this is called the flux weighted method. It is clear however that (for many well known reasons) using such group constants generally leads to unacceptable discrepancies in the reaction rates and consequently in the multiplication factors, in particular when absorbing or strong moderating regions are present. Despite those difficulties, if we want to perform diffusion calculations, we must introduce effective constants that reproduce transport reaction rates. The diffusion calculation with these constants will be equivalent, in a certain sense, to a reference. To achieve this goal, a code (EQUIPAGE) was written. Using an iterative procedure, matching reaction rates, this code is able to generate for each of the broad-groups a set of equivalent constants to be used in a finite difference diffusion calculation

252

Investigation of Calculation Techniques of Finite Difference Method

Directory of Open Access Journals (Sweden)

Full Text Available Finite difference method used for microstrip transmission line analysis is considered in this article. Paper mainly deals with iterative and bound matrix calculation techniques of finite difference method. Mathematical model for microstrip transmission line electrical potential calculations using both techniques is described. Results of characteristic impedance calculation using iterative and bound matrix techniques are presented and analyzed.Article in Lithuanian

Audrius Krukonis

2011-03-01

253

A numerical study of the steady, axisymmetric flow in a heated, rotating spherical shell is conducted to model the Atmospheric General Circulation Experiment (AGCE) proposed to run aboard a later Shuttle mission. The AGCE will consist of concentric rotating spheres confining a dielectric fluid. By imposing a dielectric field across the fluid a radial body force will be created. The numerical solution technique is based on the incompressible Navier-Stokes equations. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This paper discusses the development and performance of this numerical scheme for the AGCE which has been modeled in the past only by pure FD formulations. In addition, previous models have not investigated the effect of using a dielectric force to simulate terrestrial gravity. The effect of this dielectric force on the flow field is investigated as well as a parameter study of varying rotation rates and boundary temperatures. Among the effects noted are the production of larger velocities and enhanced reversals of radial temperature gradients for a body force generated by the electric field.

Macaraeg, M. G.

1985-01-01

254

Discrete earth models are commonly represented by uniform structured grids. In order to ensure accurate numerical description of all wave components propagating through these uniform grids, the grid size must be determined by the slowest velocity of the entire model. Consequently, high velocity areas are always oversampled, which inevitably increases the computational cost. A practical solution to this problem is to use nonuniform grids. We propose a nonuniform grid implicit spatial finite difference method which utilizes nonuniform grids to obtain high efficiency and relies on implicit operators to achieve high accuracy. We present a simple way of deriving implicit finite difference operators of arbitrary stencil widths on general nonuniform grids for the first and second derivatives and, as a demonstration example, apply these operators to the pseudo-acoustic wave equation in tilted transversely isotropic (TTI) media. We propose an efficient gridding algorithm that can be used to convert uniformly sampled models onto vertically nonuniform grids. We use a 2D TTI salt model to demonstrate its effectiveness and show that the nonuniform grid implicit spatial finite difference method can produce highly accurate seismic modeling results with enhanced efficiency, compared to uniform grid explicit finite difference implementations.

Chu, Chunlei; Stoffa, Paul L.

2012-01-01

255

International Nuclear Information System (INIS)

This report presents comparisons of results of five implicit and explicit finite difference recession computation techniques with results from a more accurate ''benchmark'' solution applied to a simple one-dimensional nonlinear ablation problem. In the comparison problem a semi-infinite solid is subjected to a constant heat flux at its surface and the rate of recession is controlled by the solid material's latent heat of fusion. All thermal properties are assumed constant. The five finite difference methods include three front node dropping schemes, a back node dropping scheme, and a method in which the ablation problem is embedded in an inverse heat conduction problem and no nodes are dropped. Constancy of thermal properties and the semiinfinite and one-dimensional nature of the problem at hand are not necessary assumptions in applying the methods studied to more general problems. The best of the methods studied will be incorporated into APL's Standard Heat Transfer Program

256

Digital Repository Infrastructure Vision for European Research (DRIVER)

In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is 2M when we use 2nd order time dom...

Ghorai, Anjana P.; Tiwary, R.

2013-01-01

257

Eigenvalues of singular differential operators by finite difference methods. II.

Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.

Baxley, J. V.

1972-01-01

258

Discrete Mechanics and Optimal Control: an Analysis

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationsh...

Ober-Bloebaum, S; Marsden, J E

2008-01-01

259

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier-Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier-Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity-stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.

Einkemmer, L.; Wiesenberger, M.

2014-11-01

260

Digital Repository Infrastructure Vision for European Research (DRIVER)

In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second-order accurate convex splitting scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The fully discrete, fully second-order finite difference scheme in question was derived in a recent work [2]. An introduction of a new variable \\psi, corresponding to the tempor...

Baskaran, Arvind; Lowengrub, John; Wang, Cheng; Wise, Steve

2012-01-01

261

Stability analysis of single-phase thermosyphon loops by finite difference numerical methods

International Nuclear Information System (INIS)

In this paper, examples of the application of finite difference numerical methods in the analysis of stability of single-phase natural circulation loops are reported. The problem is here addressed for its relevance for thermal-hydraulic system code applications, in the aim to point out the effect of truncation error on stability prediction. The methodology adopted for analysing in a systematic way the effect of various finite difference discretization can be considered the numerical analogue of the usual techniques adopted for PDE stability analysis. Three different single-phase loop configurations are considered involving various kinds of boundary conditions. In one of these cases, an original dimensionless form of the governing equations is proposed, adopting the Reynolds number as a flow variable. This allows for an appropriate consideration of transition between laminar and turbulent regimes, which is not possible with other dimensionless forms, thus enlarging the field of validity of model assumptions. (author). 14 refs., 8 figs

262

Gain from a mixed finite-difference formulation for three-dimensional diffusion-theory neutronics

International Nuclear Information System (INIS)

The advantage of a mixed differencing scheme for representing the diffusion theory approximation to neutron transport in three-dimensional triangular-Z geometry is demonstrated for a fast reactor. Most of the early codes employed the mesh edge difference formulation as is used in the German D3E code. A mesh centered formulation was chosen for use on a routine basis with mesh points located at the centers of the finite difference elements instead of at the corners where the internal material interfaces intersect, the VENURE code being the latest to use this scheme. Results are presented for a fast reactor core problem modeling hexagonal assemblies

263

Finite difference modelling of rupture propagation with strong velocity-weakening friction

We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time steps.

Rojas, Otilio; Dunham, Eric M.; Day, Steven M.; Dalguer, Luis A.; Castillo, Jose E.

2009-12-01

264

The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.

Ghil, M.; Balgovind, R.

1979-01-01

265

An assessment of semi-discrete central schemes for hyperbolic conservation laws

International Nuclear Information System (INIS)

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 clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit

266

An analytical discrete ordinates solution for two-dimensional problems based on nodal schemes

International Nuclear Information System (INIS)

In this work, the ADO method is used to solve the integrated one dimensional equations generated by the application of a nodal scheme on the two dimensional transport problem in cartesian geometry. Particularly, relations between the averaged fluxes and the unknown fluxes at the boundary are introduced as the usually needed auxiliary equations. The ADO approach, along with a level symmetric quadrature scheme, lead to an important reduction in the order of the associated eigenvalue systems. Numerical results are presented for a two dimensional problem in order to compare with available results in the literature. (author)

267

Discrete memory schemes for finite strain thermoplasticity and application to shape memory alloys

International Nuclear Information System (INIS)

A theory of finite strain plasticity has been proposed: The scheme of pure hysteresis with mixed transport has been extended to the case of non-rotational kinematics. Secondly, the simple shear case has been studied, taking into account Drucker's recent analysis regarding the 'appropriate simple idealizations for finite plasticity'. Illustrations are provided for general stress/strain paths. Also a new theory of isotropic hyperelasticity has been proposed. The 'reversible' relative Cauchy stress tensor (of type (1,1) and weight one) is defined in the dragged along coordinates as a tensorial isotropic function of the Almansi tensor and of its invariants (through the partial derivatives of the actual scalar density of elastic energy per unit extent of dragged along coordinates). The correspondance between strain and stress paths is then defined in a general form which is particularly convenient for the study of first order effects, limit behaviours, coupling and second order effects. Illustrations are provided. The addition of the pure hysteresis stress contribution ?a and of the reversible contribution ?rev leads to a scheme of 'superelasticity' departure to obtain a provisional scheme of shape memory effects. Some remarks are given regarding some of the possible generalizations of the scheme. (orig./GL)

268

An assessment of semi-discrete central schemes for hyperbolic conservation laws.

Energy Technology Data Exchange (ETDEWEB)

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 clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The 'well-designed' integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit.

Christon, Mark Allen; Robinson, Allen Conrad; Ketcheson, David Isaac

2003-09-01

269

High-order compact difference scheme for the numerical solution of time fractional heat equations.

A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space derivative. The spectral stability and the Fourier stability analysis of the difference scheme are shown. Finally a detailed numerical analysis, including tables, figures, and error comparison, is given to demonstrate the theoretical results and high accuracy of the proposed scheme. PMID:24696040

Karatay, Ibrahim; Bayramoglu, Serife R

2014-01-01

270

Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation

Energy Technology Data Exchange (ETDEWEB)

Wave propagation phenomena are important in many DOE applications such as nuclear explosion monitoring, geophysical exploration, estimating ground motion hazards and damage due to earthquakes, non-destructive testing, underground facilities detection, and acoustic noise propagation. There are also future applications that would benefit from simulating wave propagation, such as geothermal energy applications and monitoring sites for carbon storage via seismic reflection techniques. In acoustics and seismology, it is of great interest to increase the frequency bandwidth in simulations. In seismic exploration, greater frequency resolution enables shorter wave lengths to be included in the simulations, allowing for better resolution in the seismic imaging. In nuclear explosion monitoring, higher frequency seismic waves are essential for accurate discrimination between explosions and earthquakes. When simulating earthquake induced motion of large structures, such as nuclear power plants or dams, increased frequency resolution is essential for realistic damage predictions. Another example is simulations of micro-seismic activity near geothermal energy plants. Here, hydro-fracturing induces many small earthquakes and the time scale of each event is proportional to the square root of the moment magnitude. As a result, the motion is dominated by higher frequencies for smaller seismic events. The above wave propagation problems are all governed by systems of hyperbolic partial differential equations in second order differential form, i.e., they contain second order partial derivatives of the dependent variables. Our general research theme in this project has been to develop numerical methods that directly discretize the wave equations in second order differential form. The obvious advantage of working with hyperbolic systems in second order differential form, as opposed to rewriting them as first order hyperbolic systems, is that the number of differential equations in the second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a low order method.

Petersson, N A; Sjogreen, B

2012-03-26

271

Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation

International Nuclear Information System (INIS)

Wave propagation phenomena are important in many DOE applications such as nuclear explosion monitoring, geophysical exploration, estimating ground motion hazards and damage due to earthquakes, non-destructive testing, underground facilities detection, and acoustic noise propagation. There are also future applications that would benefit from simulating wave propagation, such as geothermal energy applications and monitoring sites for carbon storage via seismic reflection techniques. In acoustics and seismology, it is of great interest to increase the frequency bandwidth in simulations. In seismic exploration, greater frequency resolution enables shorter wave lengths to be included in the simulations, allowing for better resolution in the seismic imaging. In nuclear explosion monitoring, higher frequency seismic waves are essential for accurate discrimination between explosions and earthquakes. When simulating earthquake induced motion of large structures, such as nuclear power plants or dams, increased frequency resolution is essential for realistic damage predictions. Another example is simulations of micro-seismic activity near geothermal energy plants. Here, hydro-fracturing induces many small earthquakes and the time scale of each event is proportional to the square root of the moment magnitude. As a result, the motion is dominated by higher frequencies for smaller seismic events. The above wave propagation problems are all governed by systems of hyperbolic partial governed by systems of hyperbolic partial differential equations in second order differential form, i.e., they contain second order partial derivatives of the dependent variables. Our general research theme in this project has been to develop numerical methods that directly discretize the wave equations in second order differential form. The obvious advantage of working with hyperbolic systems in second order differential form, as opposed to rewriting them as first order hyperbolic systems, is that the number of differential equations in the second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a low order method.

272

The analysis of a discrete scheme of the iteratively regularized Gauss-Newton method

In this paper, the iteratively regularized Gauss-Newton method is applied to compute the stable solutions of nonlinear inverse problems. For the numerical realization, the discretization of this method is considered and the iterative solution is used to approximate the exact solution. An a posteriori rule is suggested to choose the stopping index of iteration and the convergence and rates of convergence are also derived under certain conditions. An example from the parameter identification of a differential equation problem is given to illustrate the required conditions.

Jin, Qi-Nian

2000-10-01

273

A Scheme to Share Information via Employing Discrete Algorithm to Quantum States

International Nuclear Information System (INIS)

We propose a protocol for information sharing between two legitimate parties (Bob and Alice) via public-key cryptography. In particular, we specialize the protocol by employing discrete algorithm under mod that maps integers to quantum states via photon rotations. Based on this algorithm, we find that the protocol is secure under various classes of attacks. Specially, owe to the algorithm, the security of the classical privacy contained in the quantum public-key and the corresponding cipher text is guaranteed. And the protocol is robust against the impersonation attack and the active wiretapping attack by designing particular checking processing, thus the protocol is valid. (general)

274

Eigenvalues of singular differential operators by finite difference methods. I.

Approximation of the eigenvalues of certain self-adjoint operators defined by a formal differential operator in a Hilbert space. In general, two problems are studied. The first is the problem of defining a suitable Hilbert space operator that has eigenvalues. The second problem concerns the finite difference operators to be used.

Baxley, J. V.

1972-01-01

275

On the spectrum of relativistic Schroedinger equation in finite differences

Digital Repository Infrastructure Vision for European Research (DRIVER)

We develop a method for constructing asymptotic solutions of finite-difference equations and implement it to a relativistic Schroedinger equation which describes motion of a selfgravitating spherically symmetric dust shell. Exact mass spectrum of black hole formed due to the collapse of the shell is determined from the analysis of asymptotic solutions of the equation.

Berezin, V. A.; Boyarsky, A. M.; Neronov, A. Yu

1999-01-01

276

International Nuclear Information System (INIS)

A code is presented for the numerical solution of the Boussinesq equations by means of finite differences. To deal with general complex geometries non orthogonal boundary fitted coordinates are used, which allow an arbitrary choice of the coordinate lines. It does not yield the loss of accuracy, inherent in classical finite difference schemes. Boundary conditions were examined in detail for velocity and temperature. The report describes two first applications with or without heat transfer: the flow in a cooling-tower (Navier-Stokes) and the flow in a pool of a fast breeder (Boussinesq with natural convection)

277

An Image Hiding Scheme Using 3D Sawtooth Map and Discrete Wavelet Transform

Directory of Open Access Journals (Sweden)

Full Text Available 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 domain. Experimental results show that the stego-image looks visually identical to the original host one and the secret image can be effectively extracted upon image processing attacks, which demonstrates strong robustness against a variety of attacks.

Ruisong Ye

2012-07-01

278

An outgoing energy flux boundary condition for finite difference ICRP antenna models

International Nuclear Information System (INIS)

For antennas at the ion cyclotron range of frequencies (ICRF) modeling in vacuum can now be carried out to a high level of detail such that shaping of the current straps, isolating septa, and discrete Faraday shield structures can be included. An efficient approach would be to solve for the fields in the vacuum region near the antenna in three dimensions by finite methods and to match this solution at the plasma-vacuum interface to a solution obtained in the plasma region in one dimension by Fourier methods. This approach has been difficult to carry out because boundary conditions must be imposed at the edge of the finite difference grid on a point-by-point basis, whereas the condition for outgoing energy flux into the plasma is known only in terms of the Fourier transform of the plasma fields. A technique is presented by which a boundary condition can be imposed on the computational grid of a three-dimensional finite difference, or finite element, code by constraining the discrete Fourier transform of the fields at the boundary points to satisfy an outgoing energy flux condition appropriate for the plasma. The boundary condition at a specific grid point appears as a coupling to other grid points on the boundary, with weighting determined by a kemel calctdated from the plasma surface impedance matrix for the various plasma Fourier modes. This boundary condition has been implemented in a finite difference solution of a simple problem in two dimensions, which can also be solved directly by Fourier transformation. Results are presented, and it is shown that the proposed boundary condition does enforce outgoing energy flux and yields the same solution as is obtained by Fourier methods

279

A Comparison of Time Discretization Schemes for Two-Timescale Problems in Geophysical Fluid Dynamics

In many geophysical fluid modeling applications there exist two very different time scales, essentially fast waves and slow vortices. At the largest planetary scales inertial-gravity waves propagate through the fluid at phase speeds much faster than particle velocities, while at small scales fast acoustic modes coexist with much slower vortex motions. If numerical models are to be affordable in this context, schemes which do not explicitly resolve the fast motion must be devised. Although this has traditionally been done for strictly numerical reasons, the result is physically equivalent to a temporal sub-grid-scale model. Examples of such schemes examined here are the semi-implicit, split explicit, and linear-propagator methods. All of these methods fail to resolve the fast motion accurately, but in very different ways. Recent progress in the field of rotating stratified turbulence is employed on a simplified two-time-scale version of the Boussinesq equations in order to determine how well these schemes represent the evolution of the slow variables. It is found that the split explicit and linear-propagator methods exhibit spurious interactions between the slow modes and their numerically retarded fast modes in certain conditions when ?? t is large. The semi-implicit method is able to maintain a frequency separation, but it mimics the true evolution best when time filtering is applied.

Bartello, P.

2002-06-01

280

We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils. PMID:24688360

Mattila, Keijo Kalervo; Hegele Júnior, Luiz Adolfo; Philippi, Paulo Cesar

2014-01-01

281

International Nuclear Information System (INIS)

In this paper, a H-infinity fault detection and diagnosis (FDD) scheme for a class of discrete nonlinear system fault using output probability density estimation is presented. Unlike classical FDD problems, the measured output of the system is viewed as a stochastic process and its square root probability density function (PDF) is modeled with B-spline functions, which leads to a deterministic space-time dynamic model including nonlinearities, uncertainties. A weighting mean value is given as an integral function of the square root PDF along space direction, which leads a function only about time and can be used to construct residual signal. Thus, the classical nonlinear filter approach can be used to detect and diagnose the fault in system. A feasible detection criterion is obtained at first, and a new H-infinity adaptive fault diagnosis algorithm is further investigated to estimate the fault. Simulation example is given to demonstrate the effectiveness of the proposed approaches.

282

An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine-Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating RBC2p-1 schemes (subscript denotes the accuracy order) at orders 2p and 2p+1. Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for RBC5 and RBC7 schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles.

Lerat, Alain

2014-09-01

283

Hybrid discretization of convective terms for aeroacoustics

A high order finite difference solver is implemented in order to test the accuracy and effectiveness of several numerical schemes for the aeroacoustic Large Eddy Simulations of compressible flows. The sharp gradients that are present in compressible flows and the low-dissipation required for aeroacoustics can impose contradictory requirements for the discretization of the convective terms. The present solver uses multiple discretization strategies for the convective terms such as the Roe scheme, the Kurganov-Tadmor scheme or the explicit 4-th order centered difference. Variable reconstruction is done via the 3-rd order MUSCL, with multiple limiters. A new model that blends the centered discretization with an upwind scheme tries to reconcile the contradictory requirements. The blending parameter is defined as a continuous function based on the variation of the gradient of the density field. The diffusive terms are discretized using the explicit 4-th order centered difference. The solver is parallelized for distributed memory platforms using domain decomposition and Message Passing Interface.

Cojocaru, M. G.

2013-10-01

284

Novel Finite Difference Lattice Boltzmann Model for Gas-Liquid Flow

A novel model for the finite difference lattice Boltzmann method is proposed to simulate gas-liquid two-phase flow. The effect of the large density difference is incorporated by applying acceleration modification which is deduced from macroscopic dynamics in this model. Compressibility of the fluid is adjustable. Surface tension effects are included by introducing a body force term based on the Continuum Surface Force (CSF) method. The recoloring step is replaced by a anti-diffusion scheme. Here we present results for liquid column collapse and droplet splashing on a thin liquid film. Results are in good agreement with experiment.

Wu, Long; Tsutahara, Michihisa; Tajiri, Shinsuke

2008-02-01

285

Staircase-free finite-difference time-domain formulation for general materials in complex geometries

DEFF Research Database (Denmark)

A stable Cartesian grid staircase-free finite-difference time-domain formulation for arbitrary material distributions in general geometries is introduced. It is shown that the method exhibits higher accuracy than the classical Yee scheme for complex geometries since the computational representation of physical structures is not of a staircased nature, Furthermore, electromagnetic boundary conditions are correctly enforced. The method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry. Both perfect electric conductors and dielectric structures have been investigated, Numerical results are presented and discussed.

Dridi, Kim

2001-01-01

286

Flux vector splitting of the inviscid equations with application to finite difference methods

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

Steger, J. L.; Warming, R. F.

1979-01-01

287

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

Steger, J. L.; Warming, R. F.

1981-01-01

288

Directory of Open Access Journals (Sweden)

Full Text Available Watermarking represents a potentially effective tool for the protection and verification of ownership rights in remote sensing images. Multispectral images (MSIs are the main type of images acquired by remote sensing radiometers. In this paper, a robust multispectral image watermarking technique based on the discrete wavelet transform (DWT and the tucker decomposition (TD is proposed. The core idea behind our proposed technique is to apply TD on the DWT coefficients of spectral bands of multispectral images. We use DWT to effectively separate multispectral images into different sub-images and TD to efficiently compact the energy of sub-images. Then watermark is embedded in the elements of the last frontal slices of the core tensor with the smallest absolute value. The core tensor has a good stability and represents the multispectral image properties. The experimental results on LANDSAT images show the proposed approach is robust against various types of attacks such as lossy compression, cropping, addition of noise etc.

Hai Fang

2013-11-01

289

A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.

Gladden, Herbert J.; Ko, Ching L.; Boddy, Douglas E.

1995-01-01

290

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However, the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap, combined with variable grid-size and time-step, this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

Liu, Xinxin; Yin, Xingyao; Wu, Guochen

2014-04-01

291

Digital Repository Infrastructure Vision for European Research (DRIVER)

Finite difference method (FDM) was applied to simulate thermal stress recently, which normally needs a long computational time and big computer storage. This study presents two techniques for improving computational speed in numerical simulation of casting thermal stress based on FDM, one for handling of nonconstant material properties and the other for dealing with the various coefficients in discretization equations. The use of the two techniques has been discussed and an application in wav...

Xue Xiang; Wang Yueping

2013-01-01

292

Digital Repository Infrastructure Vision for European Research (DRIVER)

We present a 2-D multi-block method for earthquake rupture dynamics in complex geometries using summation-byparts (SBP) high-order finite differences on structured grids coupled to nite volume methods on unstructured meshes. The node-centered nite volume method is used on unstructured triangular meshes to resolve earthquake ruptures propagating along non-planar faults with complex geometrical features. The unstructured meshes discretize the fault geometry only in the vicinity of the faults an...

O´reilly, Ossian; Dunham, Eric M.; Kozdon, Jeremy E.; Nordstro?m, Jan

2012-01-01

293

3D electromagnetic modeling using staggered finite differences

Energy Technology Data Exchange (ETDEWEB)

The method of finite differences has been employed to solve a variety of 3D electromagnetic (EM) forward problems arising in geophysical applications. Specific sources considered include dipolar and magnetotelluric (MT) field excitation in the frequency domain. In the forward problem, the EM fields are simulated using a vector Helmholtz equation for the electric field, which are approximated using finite differences on a staggered grid. To obtain the fields, a complex-symmetric matrix system of equations is assembled and iteratively solved using the quasi minimum method (QMR) method. Perfectly matched layer (PML) absorbing boundary conditions are included in the solution and are necessary to accurately simulate fields in propagation regime (frequencies > 10 MHZ). For frequencies approaching the static limit (< 10 KHz), the solution also includes a static-divergence correction, which is necessary to accurately simulate MT source fields and can be used to accelerate convergence for the dipolar source problem.

Newman, G.A.; Alumbaugh, D.L.

1997-06-01

294

Time-dependent optimal heater control using finite difference method

International Nuclear Information System (INIS)

Thermoforming is one of the most versatile and economical process to produce polymer products. The drawback of thermoforming is difficult to control thickness of final products. Temperature distribution affects the thickness distribution of final products, but temperature difference between surface and center of sheet is difficult to decrease because of low thermal conductivity of ABS material. In order to decrease temperature difference between surface and center, heating profile must be expressed as exponential function form. In this study, Finite Difference Method was used to find out the coefficients of optimal heating profiles. Through investigation, the optimal results using Finite Difference Method show that temperature difference between surface and center of sheet can be remarkably minimized with satisfying temperature of forming window

295

Discrete Wavelet Transform Method: A New Optimized Robust Digital Image Watermarking Scheme

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, a wavelet-based logo watermarking scheme is presented. The logo watermark is embedded into all sub-blocks of the LLn sub-band of the transformed host image, using quantization technique. Extracted logos from all sub-blocks are mixed to make the extracted watermark from distorted watermarked image. Knowing the quantization step-size, dimensions of logo and the level of wavelet transform, the watermark is extracted, without any need to have access to the original image. Robustness of the proposed algorithm was tested against the following attacks: JPEG2000 and old JPEG compression, adding salt and pepper noise, median filtering, rotating, cropping and scaling. The promising experimental results are reported and discussed.

Hassan Talebi

2012-11-01

296

Using finite difference method to simulate casting thermal stress

Digital Repository Infrastructure Vision for European Research (DRIVER)

Thermal stress simulation can provide a scientific reference to eliminate defects such as crack, residual stress centralization and deformation etc., caused by thermal stress during casting solidification. To study the thermal stress distribution during casting process, a unilateral thermal-stress coupling model was employed to simulate 3D casting stress using Finite Difference Method (FDM), namely all the traditional thermal-elastic-plastic equations are numerically and differentially discre...

Liao Dunming; Zhang Bin; Zhou Jianxin

2011-01-01

297

Finite-difference modeling experiments for seismic interferometry :

Digital Repository Infrastructure Vision for European Research (DRIVER)

In passive seismic interferometry, new reflection data can be retrieved by crosscorrelating recorded noise data. The quality of the retrieved reflection data is, among others, dependent on the duration and number of passive sources present during the recording time, the source distribution, and the source strength. To investigate these relations we set up several numerical modeling studies. To carry out the modeling in a feasible time, we design a finite-difference algorithm for the simulatio...

Thorbecke, J. W.; Draganov, D.

2011-01-01

298

Finite Difference Time Domain Method For Grating Structures

Digital Repository Infrastructure Vision for European Research (DRIVER)

The aim of this chapter is to present the principle of the FDTD method when applied to the resolution of Maxwell equations. Centered finite differences are used to approximate the value of both time and space derivatives that appear in these equations. The convergence criteria in addition to the boundary conditions (periodic or absorbing ones) are given. The special case of bi-periodic structures illuminated at oblique incidence is solved with the SFM (split field method) technique. In all ca...

Baida, F. I.; Belkhir, A.

2012-01-01

299

Some remarks on finite-difference equations of physical interest

International Nuclear Information System (INIS)

In this paper, the properties of mixed finite-difference and differential equations arising in various physical problems are investigated. First, a prescription for the correct transition to the continuum limit is given and illustrated in some concrete cases. It is further found that, by introducing a suitable generating function associated with a mixed equation, Cauchy's problem for a class of partial differential operators can be solved in a quite simple way. (author)

300

Least-Squares Image Resizing Using Finite Differences

Digital Repository Infrastructure Vision for European Research (DRIVER)

We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree n. A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a. For a given choice of basis functio...

Mun?oz Barrutia, A.; Blu, T.; Unser, M.

2001-01-01

301

Stochastic finite differences for elliptic diffusion equations in stratified domains

Digital Repository Infrastructure Vision for European Research (DRIVER)

We describe Monte Carlo algorithms to solve elliptic partial differen- tial equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess an higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or...

Maire, Sylvain; Nguyen, Giang

2013-01-01

302

A Skin Tone Based Stenographic Scheme using Double Density Discrete Wavelet Transforms.

Directory of Open Access Journals (Sweden)

Full Text Available Steganography is the art of concealing the existence of data in another transmission medium i.e. image, audio, video files to achieve secret communication. It does not replace cryptography but rather boosts the security using its obscurity features. In the proposed method Biometric feature (Skin tone region is used to implement Steganography[1]. In our proposed method Instead of embedding secret data anywhere in image, it will be embedded in only skin tone region. This skin region provides excellent secure location for data hiding. So, firstly skin detection is performed using, HSV (Hue, Saturation, Value color space in cover images. Thereafter, a region from skin detected area is selected, which is known as the cropped region. In this cropped region secret message is embedded using DD-DWT (Double Density Discrete Wavelet Transform. DD-DWT overcomes the intertwined shortcomings of DWT (like poor directional selectivity, Shift invariance, oscillations and aliasing[2].optimal pixel adjustment process (OPA is used to enhance the image quality of the stego-image. Hence the image obtained after embedding secret message (i.e. Stego image is far more secure and has an acceptable range of PSNR. The proposed method is much better than the previous works both in terms of PSNR and robustness against various attacks (like Gaussian Noise, salt and pepper Noise, Speckle Noise, rotation, JPEG Compression, Cropping, and Contrast Adjustment etc.

Varsha Gupta

2013-07-01

303

Finite difference modeling of seismic wave propagation in monoclinic media

Reported in the present paper are the results of the study of propagation of SH waves in the plane of mirror symmetry of a monoclinic multilayered medium with displacement normal to the plane. Dispersion equation has been obtained analytically ussing Haskell’s matrix method, while the finite-difference method has been employed to model the SH-wave propagation to study its phase and group velocities. The stability analysis has been carried out to minimize the exponential growth of the error of finite difference approximation in order to make the finite difference method stable and convergent. Further, variations of phase velocity with respect to both wave number and dispersion parameter for different stability ratios in monoclinic media have been examined and shown graphically. The effect of change of stability ratio on the group velocity of the wave propagation has been also investigated. Likewise, the effects of change of dispersion parameter on phase velocity and the variation of frequency with increase of wave number have been graphically represented and discussed.

Kalyani, Vijay Kumar; Sinha, Amalendu; Pallavika; Chakraborty, Swapan Kumar; Mahanti, N. C.

2008-12-01

304

Introduction to finite-difference methods for numerical fluid dynamics

Energy Technology Data Exchange (ETDEWEB)

This work is intended to be a beginner`s exercise book for the study of basic finite-difference techniques in computational fluid dynamics. It is written for a student level ranging from high-school senior to university senior. Equations are derived from basic principles using algebra. Some discussion of partial-differential equations is included, but knowledge of calculus is not essential. The student is expected, however, to have some familiarity with the FORTRAN computer language, as the syntax of the computer codes themselves is not discussed. Topics examined in this work include: one-dimensional heat flow, one-dimensional compressible fluid flow, two-dimensional compressible fluid flow, and two-dimensional incompressible fluid flow with additions of the equations of heat flow and the {Kappa}-{epsilon} model for turbulence transport. Emphasis is placed on numerical instabilities and methods by which they can be avoided, techniques that can be used to evaluate the accuracy of finite-difference approximations, and the writing of the finite-difference codes themselves. Concepts introduced in this work include: flux and conservation, implicit and explicit methods, Lagrangian and Eulerian methods, shocks and rarefactions, donor-cell and cell-centered advective fluxes, compressible and incompressible fluids, the Boussinesq approximation for heat flow, Cartesian tensor notation, the Boussinesq approximation for the Reynolds stress tensor, and the modeling of transport equations. A glossary is provided which defines these and other terms.

Scannapieco, E.; Harlow, F.H.

1995-09-01

305

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree-type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress-conservative finite difference constraints which are enforced at the "hanging" velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.

Gerya, T. V.; May, D. A.; Duretz, T.

2013-04-01

306

A finite difference method for the design of gradient coils in MRI--an initial framework.

This paper proposes a finite-difference (FD)-based method for the design of gradient coils in MRI. The design method first uses the FD approximation to describe the continuous current density of the coil space and then employs the stream function method to extract the coil patterns. During the numerical implementation, a linear equation is constructed and solved using a regularization scheme. The algorithm details have been exemplified through biplanar and cylindrical gradient coil design examples. The design method can be applied to unusual coil designs such as ultrashort or dedicated gradient coils. The proposed gradient coil design scheme can be integrated into a FD-based electromagnetic framework, which can then provide a unified computational framework for gradient and RF design and patient-field interactions. PMID:22353392

Zhu, Minhua; Xia, Ling; Liu, Feng; Zhu, Jianfeng; Kang, Liyi; Crozier, Stuart

2012-09-01

307

Five-point form of the nodal diffusion method and comparison with finite-difference

International Nuclear Information System (INIS)

Nodal Methods have been derived, implemented and numerically tested for several problems in physics and engineering. In the field of nuclear engineering, many nodal formalisms have been used for the neutron diffusion equation, all yielding results which were far more computationally efficient than conventional Finite Difference (FD) and Finite Element (FE) methods. However, not much effort has been devoted to theoretically comparing nodal and FD methods in order to explain the very high accuracy of the former. In this summary we outline the derivation of a simple five-point form for the lowest order nodal method and compare it to the traditional five-point, edge-centered FD scheme. The effect of the observed differences on the accuracy of the respective methods is established by considering a simple test problem. It must be emphasized that the nodal five-point scheme derived here is mathematically equivalent to previously derived lowest order nodal methods. 7 refs., 1 tab

308

This project is about the development of high order, non-oscillatory type schemes for computational fluid dynamics. Algorithm analysis, implementation, and applications are performed. Collaborations with NASA scientists have been carried out to ensure that the research is relevant to NASA objectives. The combination of ENO finite difference method with spectral method in two space dimension is considered, jointly with Cai [3]. The resulting scheme behaves nicely for the two dimensional test problems with or without shocks. Jointly with Cai and Gottlieb, we have also considered one-sided filters for spectral approximations to discontinuous functions [2]. We proved theoretically the existence of filters to recover spectral accuracy up to the discontinuity. We also constructed such filters for practical calculations.

Shu, Chi-Wang

1998-01-01

309

I introduce a new formulation of the equations of geophysical fluid dynamics (GFD) consisting of sets of topological and metric covariant equations, which allows a systematic discretization by applying the tools of discrete exterior calculus (DEC). Within the covariant equations, where the form of the equations is invariant under coordinate transformation, the prognostic variables describing the evolution of the fluid are represented by differential forms. By introducing additional auxiliary prognostic variables, I split the geophysical fluid equations in a topological and in a metric part. The resulting set of equations has similar form to the covariant linear Maxwell's equations and allows to use concepts of electrodynamics also within fluid dynamics. Moreover, such formulation enables a systematic discretization according to DEC by using chains and cochains to approximate manifolds and differential forms, respectively. The discrete scheme follows from the choice of the topological meshes (chains) for the momentum and continuity equations and from the discrete representation of the metric equations (discrete Hodge-star operator, discrete interior product). I illustrate that this formulation incorporates several finite difference schemes, for instance, the triangular and the hexagonal C-grid discretization of the non-rotating linear shallow-water equations, for which consistency and stability for uniform and non-uniform grids are shown.

Bauer, Werner

2013-04-01

310

Finite difference evolution equations and quantum dynamical semigroups

International Nuclear Information System (INIS)

We consider the recently proposed [Bonifacio, Lett. Nuovo Cimento, 37, 481 (1983)] coarse grained description of time evolution for the density operator rho(t) through a finite difference equation with steps tau, and we prove that there exists a generator of the quantum dynamical semigroup type yielding an equation giving a continuous evolution coinciding at all time steps with the one induced by the coarse grained description. The map rho(0)?rho(t) derived in this way takes the standard form originally proposed by Lindblad [Comm. Math. Phys., 48, 119 (1976)], even when the map itself (and, therefore, the corresponding generator) is not bounded. (author)

311

A review of current finite difference rotor flow methods

Rotary-wing computational fluid dynamics is reaching a point where many three-dimensional, unsteady, finite-difference codes are becoming available. This paper gives a brief review of five such codes, which treat the small disturbance, conservative and nonconservative full-potential, and Euler flow models. A discussion of the methods of applying these codes to the rotor environment (including wake and trim considerations) is followed by a comparison with various available data. These data include tests of advancing lifting and nonlifting, and hovering model rotors with significant supercritical flow regions. The codes are also compared for computational efficiency.

Caradonna, F. X.; Tung, C.

1986-01-01

312

Finite difference program for calculating hydride bed wall temperature profiles

International Nuclear Information System (INIS)

A QuickBASIC finite difference program was written for calculating one dimensional temperature profiles in up to two media with flat, cylindrical, or spherical geometries. The development of the program was motivated by the need to calculate maximum temperature differences across the walls of the Tritium metal hydrides beds for thermal fatigue analysis. The purpose of this report is to document the equations and the computer program used to calculate transient wall temperatures in stainless steel hydride vessels. The development of the computer code was motivated by the need to calculate maximum temperature differences across the walls of the hydrides beds in the Tritium Facility for thermal fatigue analysis

313

Discrete conservation laws and the convergence of long time simulations of the mKdV equation

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.

Gorria, Carlos; Vega, Luis

2011-01-01

314

International Nuclear Information System (INIS)

A general adjoint Monte Carlo-forward discrete ordinates radiation transport calculational scheme has been created to study the effects of the radiation environment in Hiroshima and Nagasaki due to the bombing of these two cities. Various such studies for comparison with physical data have progressed since the end of World War II with advancements in computing machinery and computational methods. These efforts have intensified in the last several years with the U.S.-Japan joint reassessment of nuclear weapons dosimetry in Hiroshima and Nagasaki. Three principal areas of investigation are: (1) to determine by experiment and calculation the neutron and gamma-ray energy and angular spectra and total yield of the two weapons; (2) using these weapons descriptions as source terms, to compute radiation effects at several locations in the two cities for comparison with experimental data collected at various times after the bombings and thus validate the source terms; and (3) to compute radiation fields at the known locations of fatalities and surviving individuals at the time of the bombings and thus establish an absolute cause-and-effect relationship between the radiation received and the resulting injuries to these individuals and any of their descendants as indicated by their medical records. It is in connection with the second and third items, the determination of the radiation effects and the dose received by individuals, that the current study is concernede current study is concerned

315

Digital Repository Infrastructure Vision for European Research (DRIVER)

An untraditional space-time method for describing the dynamics of high-field electron-hole wave packets in semiconductor quantum wells is presented. A finite-difference time-domain technique is found to be computationally efficient and can incorporate Coulomb, static, terahertz, and magnetic fields to all orders, and thus can be applied to study many areas of high-field semiconductor physics. Several electro-optical and electro-magneto-optical excitation schemes are studied, some well know...

Hughes, S.

2004-01-01

316

Finite difference analysis of curved deep beams on Winkler foundation

Directory of Open Access Journals (Sweden)

Full Text Available This research deals with the linear elastic behavior of curved deep beams resting on elastic foundations with both compressional and frictional resistances. Timoshenko’s deep beam theory is extended to include the effect of curvature and the externally distributed moments under static conditions. As an application to the distributed moment generations, the problems of deep beams resting on elastic foundations with both compressional and frictional restraints have been investigated in detail. The finite difference method was used to represent curved deep beams and the results were compared with other methods to check the accuracy of the developed analysis. Several important parameters are incorporated in the analysis, namely, the vertical subgrade reaction, horizontal subgrade reaction, beam width, and also the effect of beam thickness to radius ratio on the deflections, bending moments, and shear forces. The computer program (CDBFDA (Curved Deep Beam Finite Difference Analysis Program coded in Fortran-77 for the analysis of curved deep beams on elastic foundations was formed. The results from this method are compared with other methods exact and numerical and check the accuracy of the solutions. Good agreements are found, the average percentages of difference for deflections and moments are 5.3% and 7.3%, respectively, which indicate the efficiency of the adopted method for analysis.

Adel A. Al-Azzawi

2011-03-01

317

Arrayed waveguide grating using the finite difference beam propagation method

The purpose of this work is to analyze by simulation the coupling effects occurring in Arrayed Waveguide Grating (AWG) using the finite difference beam propagation method (FD-BPM). Conventional FD-BPM techniques do not immediately lend themselves to the analysis of large structures such as AWG. Cooper et al.1 introduced a description of the coupling between the interface of arrayed waveguides and star couplers using the numerically-assisted coupled-mode theory. However, when the arrayed waveguides are spatially close, such that, there is strong coupling between them, and coupled-mode theory is not adequate. On the other hand, Payne2 developed an exact eigenvalue equation for the super modes of a straight arrayed waveguide which involve a computational overhead. In this work, an integration of both methods is accomplished in order to describe the behavior of the propagation of light in guided curves. This new method is expected to reduce the necessary effort for simulation while also enabling the simulation of large and curved arrayed waveguides using a fully vectorial finite difference technique.

Toledo, M. C. F.; Alayo, M. I.

2013-03-01

318

Finite-Difference Frequency-Domain Method in Nanophotonics

DEFF Research Database (Denmark)

Optics and photonics are exciting, rapidly developing fields building their success largely on use of more and more elaborate artificially made, nanostructured materials. To further advance our understanding of light-matter interactions in these complicated artificial media, numerical modeling is often indispensable. This thesis presents the development of rigorous finite-difference method, a very general tool to solve Maxwell’s equations in arbitrary geometries in three dimensions, with an emphasis on the frequency-domain formulation. Enhanced performance of the perfectly matched layers is obtained through free space squeezing technique, and nonuniform orthogonal grids are built to greatly improve the accuracy of simulations of highly heterogeneous nanostructures. Examples of the use of the finite-difference frequency-domain method in this thesis range from simulating localized modes in a three-dimensional photonic-crystal membrane-based cavity, a quasi-one-dimensional nanobeam cavity and arrays of side-coupled nanobeam cavities, to modeling light propagation through metal films with single or periodically arranged multiple subwavelength slits.

Ivinskaya, Aliaksandra

2011-01-01

319

Discrete geometric conservation laws (DGCLs) govern the geometric parameters of numerical schemes designed for the solution of unsteady flow problems on moving grids. A DGCL requires that these geometric parameters, which include among others grid positions and velocities, be computed so that the corresponding numerical scheme reproduces exactly a constant solution. Sometimes, this requirement affects the intrinsic design of an arbitrary Lagrangian Eulerian (ALE) solution method. In this paper, we show for sample ALE schemes that satisfying the corresponding DGCL is a necessary and sufficient condition for a numerical scheme to preserve the nonlinear stability of its fixed grid counterpart. We also highlight the impact of this theoretical result on practical applications of computational fluid dynamics.

Farhat, Charbel; Geuzaine, Philippe; Grandmont, Céline

2001-12-01

320

Directory of Open Access Journals (Sweden)

Full Text Available In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM. We have proved that the accuracy of this finite-difference scheme is 2M when we use 2nd order time domain finite-difference and 2M-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu et al. (2009.

Anjana P. Ghorai

2013-03-01

321

Conservative high-order-accurate finite-difference methods for curvilinear grids

Two fourth-order-accurate finite-difference methods for numerically solving hyperbolic systems of conservation equations on smooth curvilinear grids are presented. The first method uses the differential form of the conservation equations; the second method uses the integral form of the conservation equations. Modifications to these schemes, which are required near boundaries to maintain overall high-order accuracy, are discussed. An analysis that demonstrates the stability of the modified schemes is also provided. Modifications to one of the schemes to make it total variation diminishing (TVD) are also discussed. Results that demonstrate the high-order accuracy of both schemes are included in the paper. In particular, a Ringleb-flow computation demonstrates the high-order accuracy and the stability of the boundary and near-boundary procedures. A second computation of supersonic flow over a cylinder demonstrates the shock-capturing capability of the TVD methodology. An important contribution of this paper is the dear demonstration that higher order accuracy leads to increased computational efficiency.

Rai, Man M.; Chakrvarthy, Sukumar

1993-01-01

322

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments. PMID:24489518

2013-01-01

323

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments. PMID:24489518

Kudryavtsev, Oleg

2013-01-01

324

International Nuclear Information System (INIS)

Radiative heat transfer is the dominant mode of heat transfer in many engineering problems, including combustion chambers, space, greenhouses, rocket plume sensing, among others. The aim of this study is to develop an efficient method capable of eliminating ray effects in complex 2D situations and to use the developed code for other problems including combined conduction and convection in connection with CFD codes. A complete genuinely multidimensional discretization in two-dimensional discrete ordinates method is formulated to solve radiative heat transfer in a rectangular enclosure composed of diffusely emitting and reflecting boundaries and containing homogeneous media that absorbs, emits and scatters radiation. A new genuinely multidimensional differencing scheme is used to solve the radiative transfer equation with S4, S6, S8, T6, T7, T8 and T9 angular quadrature schemes. Different cases are analyzed and the results are compared when possible with those obtained by others researchers

325

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, an implicit logarithmic finite difference method (I-LFDM is implemented for the numerical solution of one dimensional coupled nonlinear Burgers’ equation. The numerical scheme provides a system of nonlinear difference equations which we linearise using Newton's method. The obtained linear system via Newton's method is solved by Gauss elimination with partial pivoting algorithm. To illustrate the accuracy and reliability of the scheme, three numerical examples are described. The obtained numerical solutions are compared well with the exact solutions and those already available.

Vineet K. Srivastava

2014-03-01

326

Finite-difference modeling of commercial aircraft using TSAR

Energy Technology Data Exchange (ETDEWEB)

Future aircraft may have systems controlled by fiber optic cables, to reduce susceptibility to electromagnetic interference. However, the digital systems associated with the fiber optic network could still experience upset due to powerful radio stations, radars, and other electromagnetic sources, with potentially serious consequences. We are modeling the electromagnetic behavior of commercial transport aircraft in support of the NASA Fly-by-Light/Power-by-Wire program, using the TSAR finite-difference time-domain code initially developed for the military. By comparing results obtained from TSAR with data taken on a Boeing 757 at the Air Force Phillips Lab., we hope to show that FDTD codes can serve as an important tool in the design and certification of U.S. commercial aircraft, helping American companies to produce safe, reliable air transportation.

Pennock, S.T.; Poggio, A.J.

1994-11-15

327

Visualization of elastic wavefields computed with a finite difference code

Energy Technology Data Exchange (ETDEWEB)

The authors have developed a finite difference elastic propagation model to simulate seismic wave propagation through geophysically complex regions. To facilitate debugging and to assist seismologists in interpreting the seismograms generated by the code, they have developed an X Windows interface that permits viewing of successive temporal snapshots of the (2D) wavefield as they are calculated. The authors present a brief video displaying the generation of seismic waves by an explosive source on a continent, which propagate to the edge of the continent then convert to two types of acoustic waves. This sample calculation was part of an effort to study the potential of offshore hydroacoustic systems to monitor seismic events occurring onshore.

Larsen, S. [Lawrence Livermore National Lab., CA (United States); Harris, D.

1994-11-15

328

Cm solutions of systems of finite difference equations

Directory of Open Access Journals (Sweden)

Full Text Available Let Ã¢Â„Â be the real number axis. Suppose that G, H are Cm maps from Ã¢Â„Â2n+3 to Ã¢Â„Â. In this note, we discuss the system of finite difference equations G(x,f(x,f(x+1,Ã¢Â€Â¦,f(x+n,g(x,g(x+1,Ã¢Â€Â¦,g(x+n+0 and H(x,g(x,g(x+1,Ã¢Â€Â¦,g(x+n,f(x,f(x+1,Ã¢Â€Â¦,f(x+n=0 for all xÃ¢ÂˆÂˆÃ¢Â„Â, and give some relatively weak conditions for the above system of equations to have unique Cm solutions (mÃ¢Â‰Â¥0.

Jianmin Ma

2003-06-01

329

Obtaining Potential Field Solution with Spherical Harmonics and Finite Differences

Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current and divergence free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: i) Remeshing the magnetogram onto a grid with uniform resolution in latitude, and limiting the highest order of the spherical harmonics to the anti-alias limit; ii) Using an iterative finite difference algorithm t...

Toth, Gabor; Huang, Zhenguang; 10.1088/0004-637X/732/2/102

2011-01-01

330

Non-finite-difference algorithm for integrating Newton's motion equations

We have presented some practical consequences on the molecular-dynamics simulations arising from the numerical algorithm published recently in paper Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference method and therefore it could be complementary to the traditional numerical integrating of the motion equations. It consists of two steps. First, an analytic form of polynomials in some formal parameter $\\lambda$ (we put $\\lambda=1$ after all) is derived, which approximate the solution of the system of differential equations under consideration. Next, the numerical values of the derived polynomials in the interval, in which the difference between them and their truncated part of smaller degree does not exceed a given accuracy $\\epsilon$, become the numerical solution. The particular examples, which we have considered, represent the forced linear and nonlinear oscillator and the 2D Lennard-Jones fluid. In the latter case we have restricted to the polynomials of the first degree in formal...

Brzostowski, B; Grabiec, B; Nadzieja, T

2007-01-01

331

The application of finite difference methods to normal resistivity logs

Energy Technology Data Exchange (ETDEWEB)

Measurements made using a multiple=spaced normal resistivity tool should be capable of yielding a resistivity profile defining the extent and nature of the invaded zone as well as giving accurate values for Rt. This paper outlines the development of a suitable tool, a digital data transmission system, and the mathematical analysis used to extract the required information. A modelling technique based on finite difference analysis is used to predict the borehole measurements in any given section of formation, taking into account both vertical and radial variations of resistivity. Optimisation techniques are then employed which operate on the model by varying resistivity profiles until the prediction agrees with the measured values. Alternative strategies, and the practical problems involved in their implementation in computer software, will also be discussed.

Yuratich, M.A.; Meger, W.J.

1984-01-01

332

Effects of sources on time-domain finite difference models.

Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed. PMID:24993210

Botts, Jonathan; Savioja, Lauri

2014-07-01

333

Computational electrodynamics the finite-difference time-domain method

This extensively revised and expanded third edition of the Artech House bestseller, Computational Electrodynamics: The Finite-Difference Time-Domain Method, offers engineers the most up-to-date and definitive resource on this critical method for solving Maxwell's equations. The method helps practitioners design antennas, wireless communications devices, high-speed digital and microwave circuits, and integrated optical devices with unsurpassed efficiency. There has been considerable advancement in FDTD computational technology over the past few years, and the third edition brings professionals the very latest details with entirely new chapters on important techniques, major updates on key topics, and new discussions on emerging areas such as nanophotonics. What's more, to supplement the third edition, the authors have created a Web site with solutions to problems, downloadable graphics and videos, and updates, making this new edition the ideal textbook on the subject as well.

Taflove, Allen

2005-01-01

334

Experiments with explicit filtering for LES using a finite-difference method

The equations for large-eddy simulation (LES) are derived formally by applying a spatial filter to the Navier-Stokes equations. The filter width as well as the details of the filter shape are free parameters in LES, and these can be used both to control the effective resolution of the simulation and to establish the relative importance of different portions of the resolved spectrum. An analogous, but less well justified, approach to filtering is more or less universally used in conjunction with LES using finite-difference methods. In this approach, the finite support provided by the computational mesh as well as the wavenumber-dependent truncation errors associated with the finite-difference operators are assumed to define the filter operation. This approach has the advantage that it is also 'automatic' in the sense that no explicit filtering: operations need to be performed. While it is certainly convenient to avoid the explicit filtering operation, there are some practical considerations associated with finite-difference methods that favor the use of an explicit filter. Foremost among these considerations is the issue of truncation error. All finite-difference approximations have an associated truncation error that increases with increasing wavenumber. These errors can be quite severe for the smallest resolved scales, and these errors will interfere with the dynamics of the small eddies if no corrective action is taken. Years of experience at CTR with a second-order finite-difference scheme for high Reynolds number LES has repeatedly indicated that truncation errors must be minimized in order to obtain acceptable simulation results. While the potential advantages of explicit filtering are rather clear, there is a significant cost associated with its implementation. In particular, explicit filtering reduces the effective resolution of the simulation compared with that afforded by the mesh. The resolution requirements for LES are usually set by the need to capture most of the energy-containing eddies, and if explicit filtering is used, the mesh must be enlarged so that these motions are passed by the filter. Given the high cost of explicit filtering, the following interesting question arises. Since the mesh must be expanded in order to perform the explicit filter, might it be better to take advantage of the increased resolution and simply perform an unfiltered simulation on the larger mesh? The cost of the two approaches is roughly the same, but the philosophy is rather different. In the filtered simulation, resolution is sacrificed in order to minimize the various forms of numerical error. In the unfiltered simulation, the errors are left intact, but they are concentrated at very small scales that could be dynamically unimportant from a LES perspective. Very little is known about this tradeoff and the objective of this work is to study this relationship in high Reynolds number channel flow simulations using a second-order finite-difference method.

Lund, T. S.; Kaltenbach, H. J.

1995-01-01

335

High-order finite difference solution for 3D nonlinear wave-structure interaction

DEFF Research Database (Denmark)

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 OceanWave3D presented in [1, 2]. A nonlinear decomposition of the solution into incident and scattered fields is used to increase the efficiency of the wave-structure interaction problem resolution. Application of the method to the diffraction of nonlinear waves around a fixed, bottom mounted circular cylinder are presented and compared to the fully nonlinear potential code XWAVE as well as to experiments.

Ducrozet, Guillaume; Bingham, Harry B.

2010-01-01

336

We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical model is first benchmarked by characterizing the rheological behaviour of dilute homogeneous solutions in various configurations, including steady shear, elongational flows, transient shear and oscillatory flows. As an upgrade of complexity, we study the model in presence of non-ideal multicomponent interfaces, where immiscibility is introduced in the LBM description using the "Shan-Chen" model. The problem of a confined viscoelastic (Newtonian) droplet in a Newtonian (viscoelastic) matrix under simple shear is investigated and numerical resu...

Gupta, A; Scagliarini, A

2014-01-01

337

Computationally efficient finite-difference modal method for the solution of Maxwell's equations.

In this work, a new implementation of the finite-difference (FD) modal method (FDMM) based on an iterative approach to calculate the eigenvalues and corresponding eigenfunctions of the Helmholtz equation is presented. Two relevant enhancements that significantly increase the speed and accuracy of the method are introduced. First of all, the solution of the complete eigenvalue problem is avoided in favor of finding only the meaningful part of eigenmodes by using iterative methods. Second, a multigrid algorithm and Richardson extrapolation are implemented. Simultaneous use of these techniques leads to an enhancement in terms of accuracy, which allows a simple method such as the FDMM with a typical three-point difference scheme to be significantly competitive with an analytical modal method. PMID:24323014

Semenikhin, Igor; Zanuccoli, Mauro

2013-12-01

338

In the finite difference method which is commonly used in computational micromagnetics, the demagnetizing field is usually computed as a convolution of the magnetization vector field with the demagnetizing tensor that describes the magnetostatic field of a cuboidal cell with constant magnetization. An analytical expression for the demagnetizing tensor is available, however at distances far from the cuboidal cell, the numerical evaluation of the analytical expression can be very inaccurate. Due to this large-distance inaccuracy numerical packages such as OOMMF compute the demagnetizing tensor using the explicit formula at distances close to the originating cell, but at distances far from the originating cell a formula based on an asymptotic expansion has to be used. In this work, we describe a method to calculate the demagnetizing field by numerical evaluation of the multidimensional integral in the demagnetization tensor terms using a sparse grid integration scheme. This method improves the accuracy of comput...

Chernyshenko, Dmitri

2014-01-01

339

Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.

Strong, Stuart L.; Meade, Andrew J., Jr.

1992-01-01

340

Black-Scholes finite difference modeling in forecasting of call warrant prices in Bursa Malaysia

Call warrant is a type of structured warrant in Bursa Malaysia. It gives the holder the right to buy the underlying share at a specified price within a limited period of time. The issuer of the structured warrants usually uses European style to exercise the call warrant on the maturity date. Warrant is very similar to an option. Usually, practitioners of the financial field use Black-Scholes model to value the option. The Black-Scholes equation is hard to solve analytically. Therefore the finite difference approach is applied to approximate the value of the call warrant prices. The central in time and central in space scheme is produced to approximate the value of the call warrant prices. It allows the warrant holder to forecast the value of the call warrant prices before the expiry date.

Mansor, Nur Jariah; Jaffar, Maheran Mohd

2014-07-01

341

A fast high-order finite difference algorithm for pricing American options

We describe an improvement of Han and Wu's algorithm [H. Han, X.Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal. 41 (6) (2003) 2081-2095] for American options. A high-order optimal compact scheme is used to discretise the transformed Black-Scholes PDE under a singularity separating framework. A more accurate free boundary location based on the smooth pasting condition and the use of a non-uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem. Extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained. Comparisons with some recently proposed methods for the American options problem are carried out to show the advantage of our numerical method.

Tangman, D. Y.; Gopaul, A.; Bhuruth, M.

2008-12-01

342

Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.

Giles, Michael B

2012-01-01

343

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model - that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable. ?? 2007 Society of Exploration Geophysicists.

Haney, M.M.

2007-01-01

344

International Nuclear Information System (INIS)

Verification that a numerical method performs as intended is an integral part of code development. Semi-analytical benchmarks enable one such verification modality. Unfortunately, a semi-analytical benchmark requires some degree of analytical forethought and treats only relatively idealized cases making it of limited diagnostic value. In the first part of our investigation (Part I, in these proceedings), we established the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrated an analytically enhanced version that leads to the analytical solution. The second part of our presentation (Part II) concerns the numerical implementation and application of the finite difference solutions of Part I. Here, we demonstrate how the numerical schemes themselves provide the semi-analytical benchmark. With the analytical solution known, we therefore have a test for accuracy of the proposed finite difference algorithms designed for high order. (authors)

345

Semi-discrete numeric solution for the non-stationary heat equation using mimetic techniques

It is proposed that the diffusion equation can be solved using second-order mimetic operators for the spatial partial derivatives, in order to obtain a semi-discrete time scheme that is easy to solve with exponential integrators. The scheme is more stable than the traditional method of finite differences (centered on space and forward on time) and easier to implement than implicit methods. Some numerical examples are shown to illustrate the advantages of the proposed method. In addition, routines written in MATLAB were developed for its implementation.

Hernández-Walls, R.; Castillo, J.; Rojas-Mayoral, E. M.; Girón-Nava, J. A.

2014-11-01

346

In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier-Stokes equations. Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation. The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge-Kutta scheme, SSPRK4.

Nigro, Alessandra; De Bartolo, Carmine; Bassi, Francesco; Ghidoni, Antonio

2014-11-01

347

3D anisotropic modeling for airborne EM systems using finite-difference method

Most current airborne EM data interpretations assume an isotropic model, which is sometimes inappropriate, especially in regions with distinct dipping anisotropy due to strong layering and stratifications. In this paper, we investigate airborne EM modeling and interpretation for a 3D earth with arbitrarily electrical anisotropy. We implement the staggered finite-difference algorithm to solve the coupled partial differential equations for the scattered electrical fields. Whereas the current density that is connected to the diagonal elements of the anisotropic conductivity tensor is discretized by using the volume weighted average, the current density that is connected to the non-diagonal elements is discretized by using the volume current density average. Further, we apply a divergence correction technique designed specifically for 3D anisotropic models to speed up the modeling process. For numerical experiments, we take both VMD and HMD transmitting dipoles for two typical anisotropic cases: 1) anisotropic anomalous inhomogeneities embedded in an isotropic half-space; and 2) isotropic anomalous inhomogeneities embedded in an anisotropic host rock. Model experiments show that our algorithm has high calculation accuracy, the divergence correction technique used in the modeling can greatly improve the convergence of the solutions, accelerating the calculation speed up to 2 times for the model presented in the paper. The characteristics inside the anisotropic earth, like the location of the anomalous body and the principal axis orientations, can also be clearly identified from AEM area surveys.

Liu, Yunhe; Yin, Changchun

2014-10-01

348

FLUOMEG: a planar finite difference mesh generator for fluid flow problems with parallel boundaries

International Nuclear Information System (INIS)

A two- or three-dimensional finite difference mesh generator capable of discretizing subrectangular flow regions (planar coordinates) with arbitrarily shaped bottom contours (vertical dimension) was developed. This economical, interactive computer code, written in FORTRAN IV and employing DISSPLA software together with graphics terminal, generates first a planar rectangular grid of variable element density according to the geometry and local kinematic flow patterns of a given fluid flow problem. Then subrectangular areas are deleted to produce canals, tributaries, bays, and the like. For three-dimensional problems, arbitrary bathymetric profiles (river beds, channel cross section, ocean shoreline profiles, etc.) are approximated with grid lines forming steps of variable spacing. Furthermore, the code works as a preprocessor numbering the discrete elements and the nodal points. Prescribed values for the principal variables can be automatically assigned to solid as well as kinematic boundaries. Cabinet drawings aid in visualizing the complete flow domain. Input data requirements are necessary only to specify the spacing between grid lines, determine land regions that have to be excluded, and to identify boundary nodes. 15 figures, 2 tables

349

Directory of Open Access Journals (Sweden)

Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z levels, s coordinates and enveloped s coordinates, all show their limitations in the areas of complex topography. Two new hybrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s coordinates or enveloped s coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations are used to identify the optimum value of Smagorinsky diffusivity coefficient to best represent the mesoscale activity.

G. Shapiro

2013-03-01

350

In 2011, Kang and Fang (Commun. Theor. Phys. 55:239-243, 2011) presented a quantum direct communication protocol using single photons. This study points out a pitfall in Kang and Fang's scheme, in which an eavesdropper can launch a measure-and-resend attack on this scheme to reveal the secret message. Furthermore, an improved scheme is proposed to avoid the attack.

Yang, Chun-Wei; Hwang, Tzonelih

2014-01-01

351

Kelvin wave propagation along straight boundaries in C-grid finite-difference models

Discrete solutions for the propagation of coastally-trapped Kelvin waves are studied, using a second-order finite-difference staggered grid formulation that is widely used in geophysical fluid dynamics (the Arakawa C-grid). The fundamental problem of linear, inviscid wave propagation along a straight coastline is examined, in a fluid of constant depth with uniform background rotation, using the shallow-water equations which model either barotropic (surface) or baroclinic (internal) Kelvin waves. When the coast is aligned with the grid, it is shown analytically that the Kelvin wave speed and horizontal structure are recovered to second-order in grid spacing h. When the coast is aligned at 45° to the grid, with the coastline approximated as a staircase following the grid, it is shown analytically that the wave speed is only recovered to first-order in h, and that the horizontal structure of the wave is infected by a thin computational boundary layer at the coastline. It is shown numerically that such first-order convergence in h is attained for all other orientations of the grid and coastline, even when the two are almost aligned so that only occasional steps are present in the numerical coastline. Such first-order convergence, despite the second-order finite differences used in the ocean interior, could degrade the accuracy of numerical simulations of dynamical phenomena in which Kelvin waves play an important role. The degradation is shown to be particularly severe for a simple example of near-resonantly forced Kelvin waves in a channel, when the energy of the forced response can be incorrect by a factor of 2 or more, even with 25 grid points per wavelength.

Griffiths, Stephen D.

2013-12-01

352

A multifunctional interface method with capabilities for variable-fidelity modeling and multiple method analysis is presented. The methodology provides an effective capability by which domains with diverse idealizations can be modeled independently to exploit the advantages of one approach over another. The multifunctional method is used to couple independently discretized subdomains, and it is used to couple the finite element and the finite difference methods. The method is based on a weighted residual variational method and is presented for two-dimensional scalar-field problems. A verification test problem and a benchmark application are presented, and the computational implications are discussed.

Ransom, Jonathan B.

2002-01-01

353

High order resolution schemes based on the NVD and TVD boundedness criteria are applied to radiative transfer problems using the DOM in two-dimensional unstructured triangular grids. The implementation of these schemes in unstructured grids requires approximations, and two implementations reported in the literature are compared with a new one. Three different methods have been used to calculate the gradient of the radiation intensity at the center of the control volumes. The various schemes are applied to several test problems, the results are compared with those obtained using the step scheme, the mean flux interpolation scheme and another high order scheme based on a truncated Taylor series expansion, and the most accurate implementations are identified. It is concluded that although the high order schemes perform much better than the others, they are not as accurate as in Cartesian coordinates, and their order of convergence is lower than in that case.

Coelho, Pedro J.

2014-08-01

354

The scattering source iterative (SI) scheme is traditionally applied to converge fine-mesh numerical solutions to fixed-source discrete ordinates (SN) neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent (low leakage). In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh scattering source, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. Therefore, we first implement a spectral nodal method that generates coarse-mesh diffusion solution that is completely free from spatial truncation errors, then we reconstruct this coarse-mesh solution within each spatial cell of the discretization grid, to further yield the initial guess for the fine-mesh scattering source in the first SN transport sweep (forward and backward) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique.

Santos, Frederico P.; Filho, Hermes Alves; Barros, Ricardo C.

2013-10-01

355

International Nuclear Information System (INIS)

The scattering source iterative (SI) scheme is traditionally applied to converge fine-mesh numerical solutions to fixed-source discrete ordinates (SN) neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent. In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh scattering source, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. Therefore, we first implement a spectral nodal method that generates coarse-mesh diffusion solution that is completely free from spatial truncation errors, then we reconstruct this coarse-mesh solution within each spatial cell of the discretization grid, to further yield the initial guess for the fine-mesh scattering source in the first SN transport sweep (?m > 0 and ?m < 0, m = 1:N) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique. (author)

356

A finite difference method of solving anisotropic scattering problems

A new method of solving radiative transfer problems is described including a comparison of its speed with that of the doubling method, and a discussion of its accuracy and suitability for computations involving variable optical properties. The method uses a discretization in angle to produce a coupled set of first-order differential equations which are integrated between discrete depth points to produce a set of recursion relations for symmetric and anti-symmetric angular sums of the radiation field at alternate depth points. The formulation given here includes depth-dependent anisotropic scattering, absorption, and internal sources, and allows arbitrary combinations of specular and non-Lambertian diffuse reflection at either or both boundaries. Numerical tests of the method show that it can return accurate emergent intensities even for large optical depths. The method is also shown to conserve flux to machine accuracy in conservative atmospheres

Barkstrom, B. R.

1976-01-01

357

Digital Repository Infrastructure Vision for European Research (DRIVER)

We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure $\\mu$. We show that the average of regular enough test functions with respect to the (possibly non unique) invariant laws of the approximations are close to the corresponding quantity for $\\mu$. More precisely, we analyze the rate of the convergence with respect to the different discretization parameters. Here we foc...

Bre?hier, Charles-edouard; Kopec, Marie

2013-01-01

358

In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter [beta]k is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87-94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401-416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561-571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645-650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523-539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599-616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN.

Andrei, Neculai

2009-08-01

359

On a finite-difference method for solving transient viscous flow problems

A method has been developed to solve the unsteady, compressible Navier-Stokes equation with the property of consistency and the ability of minimizing the equation stiffness. It relies on innovative extensions of the state-of-the-art finite-difference techniques and is composed of: (1) the upwind scheme for split-flux and the central scheme for conventional flux terms in the inviscid and viscous regions, respectively; (2) the characteristic treatment of both inviscid and viscous boundaries; (3) an ADI procedure compatible with interior and boundary points; and (4) a scalar matrix coefficient including viscous terms. The performance of this method is assessed with four sample problems; namely, a standing shock in the Laval duct, a shock reflected from the wall, the shock-induced boundary-layer separation, and a transient internal nozzle flow. The results from the present method, an existing hybrid block method, and a well-known two-step explicit method are compared and discussed. It is concluded that this method has an optimal trade-off between the solution accuracy and computational economy, and other desirable properties for analyzing transient viscous flow problems.

Li, C. P.

1983-01-01

360

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

Digital Repository Infrastructure Vision for European Research (DRIVER)

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough R0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a speci c algebraic decomposition into infective and non-infected humans and mosquitoes. Stability ...

Anguelov, Roumen; Dumont, Yves; Lubuma, Jean M. -s; Mureithi, Eunice W.

2013-01-01

361

International Nuclear Information System (INIS)

Heat pipes are being considered as a part of the thermal management system of many space crafts due to the fact that heat pipes are capable of passively transporting large amounts of thermal energy over considerable distances with essentially no temperature drop. Mathematical modeling of heat pipe performance has been developed for both transient and steady state modes: however, existing transient heat pipe models are of limited accuracy during vary rapid transients. The modeling of the response of the vapor region in the event of rapid transients (including frozen start-ups) has not been investigated fully. The purpose of this study is to model the performance of a screened wick heat pipe in rapid transient modes and to determine the limiting conditions under which the heat pipe will operate

362

Mixed convection heat transfer in two-dimensional lid-driven rectangular cavity filled with air (Pr=0.71) is studied numerically. A hybrid scheme with multiple relaxation time lattice Boltzmann method (MRT-LBM) is used to obtain the velocity field while the temperature field is deduced from energy balance equation by using the finite difference method (FDM). The main objective of this work is to investigate the model effectiveness for mixed convection flow simulation. Results are presented in terms of streamlines, isotherms and Nusselt numbers. Excellent agreement is obtained between our results and previous works. The different comparisons demonstrate the robustness and the accuracy of our proposed approach.

Bettaibi, Soufiene; Kuznik, Frédéric; Sediki, Ezeddine

2014-06-01

363

A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed. The solver is implemented in Matlab on structured meshes with variable computational domain. To verify the model, a variety of numerical test scenarios, namely, the classical one-dimensional and two-dimensional dam break, the landslide in Hong Kong in 1993 and the Nora debris flow in the Italian Alps in 2000, are executed, and the model outputs are compared with published results. It is established that the model predictions agree well with both the analytical solution as well as the field observations.

Ouyang, Chaojun; He, Siming; Xu, Qiang; Luo, Yu; Zhang, Wencheng

2013-03-01

364

Energy Technology Data Exchange (ETDEWEB)

Finite difference techniques were used to examine the coupling of radial pulsation and convection in stellar models having comparable time scales. Numerical procedures are emphasized, including diagnostics to help determine the range of free parameters.

Deupree, R.G.

1977-01-01

365

Fast finite difference Poisson solvers on heterogeneous architectures

In this paper we propose and evaluate a set of new strategies for the solution of three dimensional separable elliptic problems on CPU-GPU platforms. The numerical solution of the system of linear equations arising when discretizing those operators often represents the most time consuming part of larger simulation codes tackling a variety of physical situations. Incompressible fluid flows, electromagnetic problems, heat transfer and solid mechanic simulations are just a few examples of application areas that require efficient solution strategies for this class of problems. GPU computing has emerged as an attractive alternative to conventional CPUs for many scientific applications. High speedups over CPU implementations have been reported and this trend is expected to continue in the future with improved programming support and tighter CPU-GPU integration. These speedups by no means imply that CPU performance is no longer critical. The conventional CPU-control-GPU-compute pattern used in many applications wastes much of CPU's computational power. Our proposed parallel implementation of a classical cyclic reduction algorithm to tackle the large linear systems arising from the discretized form of the elliptic problem at hand, schedules computing on both the GPU and the CPUs in a cooperative way. The experimental result demonstrates the effectiveness of this approach.

Valero-Lara, Pedro; Pinelli, Alfredo; Prieto-Matias, Manuel

2014-04-01

366

Perfectly Matched Layer (PML) absorbing boundary conditions were first proposed by Berenger in 1994 for the Maxwell's equations of electromagnetics. Since Hu first applied the method to Euler's equations in 1996, progress made in the application of PML to Computational Aeroacoustics (CAA) includes linearized Euler equations with non-uniform mean flow, non-linear Euler equations, flows with an arbitrary mean flow direction, and non-linear clavier-Stokes equations. Although Boltzmann-BGK methods have appeared in the literature and have been shown capable of simulating aeroacoustics phenomena, very little has been done to develop absorbing boundary conditions for these methods. The purpose of this work was to extend the PML methodology to the discrete velocity Boltzmann-BGK equation (DVBE) for the case of a horizontal mean flow in two and three dimensions. The proposed extension of the PML has been accomplished in this dissertation. Both split and unsplit PML absorbing boundary conditions are presented in two and three dimensions. A finite difference and a lattice model are considered for the solution of the PML equations. The linear stability of the PML equations is investigated for both models. The small relaxation time needed for the discrete velocity Boltzmann-BC4K model to solve the Euler equations renders the explicit Runge-Kutta schemes impractical. Alternatively, implicit-explicit Runge-Kutta (IMEX) schemes are used in the finite difference model and are implemented explicitly by exploiting the special structure of the Boltzmann-BGK equation. This yields a numerically stable solution by the finite difference schemes. As the lattice model proves to be unstable, a coupled model consisting of a lattice Boltzmann (LB) method for the Ulterior domain and an IMEX finite difference method for the PML domains is proposed and investigated. Numerical examples of acoustic and vorticity waves are included to support the validity of the PML equations. In each example, accurate solutions are obtained, supporting the conclusion that PML is an effective absorbing boundary condition.

Craig, Elena

367

International Nuclear Information System (INIS)

The lowest order Nodal Integral Method (NIM) which belongs to a large class of nodal methods, the Lawrence-Dorning class, is written in a five-point, weighted-difference form and contrasted against the edge-centered Finite Difference Method (FDM). The final equations for the two methods exhibit three differences: the NIM employs almost three times as many discrete-variables (which are node- and surface-averaged values of the flux) as the FDM; the spatial weights in the NIM include hyperbolic functions opposed to the algebraic weights in the FDM; the NIM explicitly imposes continuity of the net current across cell edges. A homogeneous model problem is devised to enable an analytical study of the numerical solutions accuracy. The analysis shows that on a given mesh the FDM calculated fundamental mode eigenvalue is more accurate than that calculated by the NIM. However, the NIM calculated flux distribution is more accurate, especially when the problem size is several times as thick as the diffusion length. Numerical results for a nonhomogeneous test problem indicate the very high accuracy of the NIM for fixed source problems in such cases. 18 refs., 1 fig., 1 tab

368

Finite difference modelling of bulk high temperature superconducting cylindrical hysteresis machines

International Nuclear Information System (INIS)

A mathematical model of the critical state based on averaged fluxon motion has been implemented to solve for the current and field distributions inside a high temperature superconducting hysteresis machine. The machine consists of a rotor made from a solid cylindrical single domain HTS placed in a perpendicular rotating field. The solution technique uses the finite difference approximation for a two-dimensional domain, discretized in cylindrical polar co-ordinates. The torque generated or equivalently the hysteresis loss in such a machine has been investigated using the model. It was found that to maximize the efficiency, the field needs to penetrate the rotor such that B0/?0JcR=0.56, where B0 is the applied field amplitude, Jc is the critical current density and R is the rotor radius. This corresponds to a penetration that is 27% greater than that which reaches the centre of the rotor. An examination of the torque density distributions across the rotor reveal that for situations where the field is less than optimal, a significant increase in the performance can be achieved by removing an inner cylinder from the rotor. (author)

369

Finite difference time domain modeling of light matter interaction in light-propelled microtools

DEFF Research Database (Denmark)

Direct laser writing and other recent fabrication techniques offer a wide variety in the design of microdevices. Hence, modeling such devices requires analysis methods capable of handling arbitrary geometries. Recently, we have demonstrated the potential of microtools, optically actuated microstructures with functionalities geared towards biophotonics applications. Compared to dynamic beam shaping alone, microtools allow more complex interactions between the shaped light and the biological samples at the receiving end. For example, strongly focused light coming from a tapered tip of a microtool may trigger highly localized non linear processes in the surface of a cell. Since these functionalities are strongly dependent on design, it is important to use models that can handle complexities and take in little simplifying assumptions about the system. Hence, we use the finite difference time domain (FDTD) method which is a direct discretization of the fundamental Maxwell's equations applicable to many optical systems. Using the FDTD, we investigate light guiding through microstructures as well as the field enhancement as light comes out of our tapered wave guide designs. Such calculations save time as it helps optimize the structures prior to fabrication and experiments. In addition to field distributions, optical forces can also be obtained using the Maxwell stress tensor formulation. By calculating the forces on bent waveguides subjected to tailored static light distributions, we demonstrate novel methods of optical micromanipulation which primarily result from the particle's geometry as opposed to the directly moving the light distributions as in conventional trapping.

Bañas, Andrew Rafael; Palima, Darwin

2013-01-01

370

Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations

Directory of Open Access Journals (Sweden)

Full Text Available In this paper, a numerical solution for a system of singularly perturbed convection-diffusion equations is studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convection-diffusion equations is O(N and the relevant coefficient matrix is well conditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem. Numerical results confirm the theory of the method.

Mohammad Ghorbanzadeh

2011-07-01

371

Implicit finite-difference methods for the Euler equations

The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

Pulliam, T. H.

1985-01-01

372

Finite Difference Methods for Minimizing False Wave Dispersion in Geophysical Flow Simulations.

Finite difference (FD) algorithms were developed to minimize false wave dispersion in geophysical flow simulations. For practical simulations limited to wavelengths L >= 3.5Deltax, traditional FD schemes produce a maximum phase speed error of 46%, while the schemes developed in this dissertation limit errors to 8%. The allowable ranges on Courant number, C, and cell Reynolds number, Re_{rm c}, were greatly extended compared to standard FD schemes: O curve fitting with local, "customized" polynomials. The polynomials are "point -wise" accurate at the grid nodes in approximating time and spatial derivatives. The spatial stencil is five grid nodes, only two more than conventional, second order FD schemes. Two implicit time operators were developed: a weighted-quadratic polynomial, involving three time levels, for advective transport terms; a weighted-trapezoidal scheme for diffusive terms. Weighting for the respective operators is achieved by separate free parameters which are adjusted independently in realizing a near-optimum FD algorithm. A novel "bootstrap" procedure was fashioned to initialize the three-level time operator. Analytic, linear solutions revealed the damping and wave dispersion characteristics of the developed FD algorithms and provided a means for exploring sensitivities to the "customized" polynomial fits and free parameters in achieving near-optimum performance. Linear simulations, with the developed algorithms, compared well with the analytic solutions. The proper separation of subgrid scale (SGS) and resolvable, i.e. grid scale (GS), phenomena in nonlinear simulations was shown to require a wave cut-off filter. It was demonstrated that the process of averaging attenuates useful GS information, yet does not minimize SGS amplitudes. The nonlinear Burgers equation was computed, displaying the correct sawtooth waveform at large Re_ {rm c}. rm Re_ {rm c}>10, the addition of a wave cut-off filter is required to minimize spatial aliasing. The nonlinear solution was found to be excellent up to Re _{rm c} = 300. Higher values of Re_{rm c} would necessitate a more intricate filtering process. A preliminary study was made of Burgers equation, using the emerging discipline of symbolic computation to explore the "birth and death" of wavenumbers and to obtain an "exact," analytic solution as a benchmark for FD solutions. It is envisioned that symbolic computation will yield insights into better methods for parameterizing SGS processes in FD models.

Derickson, Russell Glynn

1992-01-01

373

The high order finite difference and multigrid methods have been successfully applied to direct numerical simulation (DNS) for flow transition in 3D channels and 3D boundary layers with 2D and 3D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semicoarsening multigrid method associated with line distributive relaxation scheme, and a new treatment of the outflow boundary condition, which needs only a very short buffer domain to damp all wave reflection, are developed. These approaches make the multigrid DNS code very accurate and efficient. This makes us not only able to do spatial DNS for the 3D channel and flat plate at low computational costs, but also able to do spatial DNS for transition in the 3D boundary layer with 3D single and multiple roughness elements. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments.

Liu, C.; Liu, Z.

1993-01-01

374

Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

International Nuclear Information System (INIS)

This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O(N) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.

375

Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

Energy Technology Data Exchange (ETDEWEB)

This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O(N) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.

Bollig, Evan F., E-mail: bollig@scs.fsu.edu [Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306 (United States); Flyer, Natasha, E-mail: flyer@ucar.edu [Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, 1850 Table Mesa Dr., Boulder, CO 80305 (United States); Erlebacher, Gordon, E-mail: gerlebacher@fsu.edu [Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL 32306 (United States)

2012-08-30

376

Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O(N) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.

Bollig, Evan F.; Flyer, Natasha; Erlebacher, Gordon

2012-08-01

377

Directory of Open Access Journals (Sweden)

Full Text Available In this article, acoustic signal based traffic flow evaluation method using the calculus of finite differences is considered. The analytical survey for traffic flow evaluations using acoustic signals is performed. The calculus of finite differences application in traffic flow evaluation is investigated. As the result of this work, the considered method of efficiency is evaluated

Posmitnyy Y. V.

2012-12-01

378

The computation of pressure waves in shock tubes by a finite difference procedure

International Nuclear Information System (INIS)

A finite difference solution of one-dimensional unsteady isentropic compressible flow equations is presented. The computer program has been tested by solving some cases of the Riemann shock tube problem. Predictions are in good agreement with those presented by other authors. Some inaccuracies may be attributed to the wave smearing consequent of the finite-difference treatment. (author)

379

Digital Repository Infrastructure Vision for European Research (DRIVER)

In this article, acoustic signal based traffic flow evaluation method using the calculus of finite differences is considered. The analytical survey for traffic flow evaluations using acoustic signals is performed. The calculus of finite differences application in traffic flow evaluation is investigated. As the result of this work, the considered method of efficiency is evaluated

Posmitnyy Y. V.; Medovschikov M. I.

2012-01-01

380

Directory of Open Access Journals (Sweden)

Full Text Available Results of a sensitivity study are presented from various configurations of the NEMO ocean model in the Black Sea. The standard choices of vertical discretization, viz. z-levels, s-coordinates and enveloped s-coordinates, all show their limitations in the areas of complex topography. Two new hydrid vertical coordinate schemes are presented: the "s-on-top-of-z" and its enveloped version. The hybrid grids use s-coordinates or enveloped s-coordinates in the upper layer, from the sea surface to the depth of the shelf break, and z-coordinates are set below this level. The study is carried out for a number of idealised and real world settings. The hybrid schemes help reduce errors generated by the standard schemes in the areas of steep topography. Results of sensitivity tests with various horizontal diffusion formulations show that the mesoscale activity is better captured with a significantly smaller value of Smagorinsky viscosity coefficient than it was previously suggested.

G. Shapiro

2012-11-01

381

International Nuclear Information System (INIS)

Transmission losses of various reactive silencers are predicted, using a time accurate finite difference method. The numerical scheme is the 3rd order upwind scheme for axisymmetric Euler equations. Main advantage of the present method is that it can simulate linear and nonlinear wave propagation phenomena in a flow field directly with minimum numerical oscillation errors. The special treatments of incident wave condition, i.e. multiple harmonics of the transparent acoustic condition are applied to the transmission loss prediction for calculation efficiency. For the validation of the present approach, circular expansion chamber silencers without mean flow and an exponential pipe with mean flow are simulated in case of linear incident wave. The computed transmission losses have quite good agreements with those of the others. The nonlinear incident wave case is also investigated to check the usefulness of this method. The periodic N wave is clearly captured without numerical oscillation errors, and the insertion losses of two different incident frequencies are compared

382

International Nuclear Information System (INIS)

A finite-difference scheme and a Galerkin scheme are compared with respect to a very accurate solution describing time-dependent advection and diffusion of air pollutants from a line source in an atmosphere vertically stratified and limited by an inversion layer. The accurate solution was achieved by applying the finite-difference scheme on a very refined grid with a very small time step. The grid size and time step were defined according to stability and accuracy criteria discussed in the text. It is found that for the problem considered the two methods can be considered equally accurate. However, the Galerkin method gives a better approximation in the vicinity of the source. This was assumed to be partly due to the different way the source term is taken into account in the two methods. Improvement of the accuracy of the finite-difference scheme was achieved by approximating, at every step, the contribution of the source term by a Gaussian puff moving and diffusing with the velocity and diffusivity of the source location, instead of utilizing a stepwise function for the numerical approximation of the delta function representing the source term

383

Poroelastic finite-difference modeling for ultrasonic waves in digital porous cores

Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with small-scale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference (FD) method of Biot's poroelastic equations, with an arbitrary even-order (2 L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer (CPML) absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultrasonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2 L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thicknesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Q sc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.

Fu, Li-Yun; Zhang, Yan; Pei, Zhenglin; Wei, Wei; Zhang, Luxin

2014-06-01

384

In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $\\sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $\\sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.

Hoogland, Jiri Kamiel; Hoogland, Jiri; Neumann, Dimitri

2000-01-01

385

Directory of Open Access Journals (Sweden)

Full Text Available The article considers data on the gross domestic product, consumer expenditures, gross investments and volume of foreign trade for the national economy. It is assumed that time is a discrete variable with one year iteration. The article uses finite-difference equations. It considers models with a high degree of the regulatory function of the state with respect to the consumer market. The econometric component is based on the hypothesis that each of the above said macro-economic indicators for this year depends on the gross domestic product for the previous time periods. Such an assumption gives a possibility to engage the least-squares method for building up linear models of the pair regression. The article obtains the time series model, which allows building point and interval forecasts for the gross domestic product for the next year based on the values of the gross domestic product for the current and previous years. The article draws a conclusion that such forecasts could be considered justified at least in the short-term prospect. From the mathematical point of view the built model is a heterogeneous finite-difference equation of the second order with constant ratios. The article describes specific features of such equations. It illustrates graphically the analytical view of solutions of the finite-difference equation. This gives grounds to differentiate national economies as sustainable growth economies, one-sided, weak or being in the stage of successful re-formation. The article conducts comparison of the listed types with specific economies of modern states.

Polshkov Yulian M.

2013-11-01

386

A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

We present a new third-order, semi-discrete, central method for approximatingsolutions to multi-dimensional systems of hyperbolic conservation laws,convection-diffusion equations, and related problems. Our method is ahigh-order extension of the recently proposed second-order, semi-discretemethod in [16]. The method is derived independently of the specific piecewise polynomialreconstruction which is based on the previously computed cell-averages. Wedemonstrate our results, by focusing on the new third-order CWENOreconstruction presented in [21]. The numerical results we present, show thedesired accuracy, high resolution and robustness of our method.

Kurganov, A; Kurganov, Alexander; Levy, Doron

2000-01-01

387

Scientific Electronic Library Online (English)

Full Text Available SciELO Colombia | Language: English Abstract in spanish En este artículo se desarrolla un nuevo esquema de cuatro puntos para la subdivisión interpolante de curvas basado en la primera derivada discreta (DFDS), el cual, reduce la formación de oscilaciones indeseables que pueden surgir en la curva límite cuando los puntos de control no obedecen a una para [...] metrización uniforme. Se empleó un conjunto de 3000 curvas cuyos puntos de control fueron generados aleatoriamente. Curvas suaves fueron obtenidas tras siete pasos de subdivisión empleando los esquemas DFDS, Cuatro-puntos (4P), Nuevo de cuatro-puntos (N4P), Cuatro-puntos ajustado (T4P) y el Esquema interpolante geométricamente controlado (GC4P). Sobre cada curva suave se evaluó la propiedad de tortuosidad. Un análisis de las distribuciones de frecuencia obtenidas para esta propiedad, empleando la prueba de Kruskal-Wallis, revela que el esquema DFDS posee los menores valores de tortuosidad en un rango más estrecho. Abstract in english This paper develops a new scheme of four points for interpolating curve subdivision based on the discrete first derivative (DFDS), which reduces the apparition of undesirable oscillations that can be formed on the limit curve when the control points do not follow a uniform parameterization. We used [...] a set of 3000 curves whose control points were randomly generated. Smooth curves were obtained after seven steps of subdivision using five schemes DFDS, Four-Point (4P), New four-point (N4P), Tight four-point (T4P) and the geometrically controlled scheme (GC4P). The tortuosity property was evaluated on every smooth curve. An analysis for the frequency distributions of this property using the Kruskal-Wallis test reveals that DFDS scheme has the lowest values in a close range.

ALBEIRO, ESPINOSA BEDOYA; GERMÁN, SÁNCHEZ TORRES; JOHN WILLIAN, BRANCH BEDOYA.

2013-08-01

388

Scientific Electronic Library Online (English)

Full Text Available SciELO Colombia | Language: English Abstract in spanish En este artículo se desarrolla un nuevo esquema de cuatro puntos para la subdivisión interpolante de curvas basado en la primera derivada discreta (DFDS), el cual, reduce la formación de oscilaciones indeseables que pueden surgir en la curva límite cuando los puntos de control no obedecen a una para [...] metrización uniforme. Se empleó un conjunto de 3000 curvas cuyos puntos de control fueron generados aleatoriamente. Curvas suaves fueron obtenidas tras siete pasos de subdivisión empleando los esquemas DFDS, Cuatro-puntos (4P), Nuevo de cuatro-puntos (N4P), Cuatro-puntos ajustado (T4P) y el Esquema interpolante geométricamente controlado (GC4P). Sobre cada curva suave se evaluó la propiedad de tortuosidad. Un análisis de las distribuciones de frecuencia obtenidas para esta propiedad, empleando la prueba de Kruskal-Wallis, revela que el esquema DFDS posee los menores valores de tortuosidad en un rango más estrecho. Abstract in english This paper develops a new scheme of four points for interpolating curve subdivision based on the discrete first derivative (DFDS), which reduces the apparition of undesirable oscillations that can be formed on the limit curve when the control points do not follow a uniform parameterization. We used [...] a set of 3000 curves whose control points were randomly generated. Smooth curves were obtained after seven steps of subdivision using five schemes DFDS, Four-Point (4P), New four-point (N4P), Tight four-point (T4P) and the geometrically controlled scheme (GC4P). The tortuosity property was evaluated on every smooth curve. An analysis for the frequency distributions of this property using the Kruskal-Wallis test reveals that DFDS scheme has the lowest values in a close range.

ALBEIRO, ESPINOSA BEDOYA; GERMÁN, SÁNCHEZ TORRES; JOHN WILLIAN, BRANCH BEDOYA.

389

We discuss a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis. We describe a semi-implicit time discretization that extends the range of numerical stability relative to an explicit scheme. We also introduce and discuss the effects of an artificial collisional operator, which is necessary to take care of the velocity space filamentation problem unavoidable in collisionless plasmas. The computational efficiency and the cost-effectiveness of this method are compared to a Particle-in-Cell (PIC) method in the case of a two-dimensional phase space. The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability, and plasma echo. The Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to the PIC.

Camporeale, Enrico; Bergen, Benjamin K; Moulton, J David

2013-01-01

390

A computational fluid dynamics code that solves the compressible Navier-Stokes equations was applied to the Taylor-Green vortex problem to examine the code s ability to accurately simulate the vortex decay and subsequent turbulence. The code, WRLES (Wave Resolving Large-Eddy Simulation), uses explicit central-differencing to compute the spatial derivatives and explicit Low Dispersion Runge-Kutta methods for the temporal discretization. The flow was first studied and characterized using Bogey & Bailley s 13-point dispersion relation preserving (DRP) scheme. The kinetic energy dissipation rate, computed both directly and from the enstrophy field, vorticity contours, and the energy spectra are examined. Results are in excellent agreement with a reference solution obtained using a spectral method and provide insight into computations of turbulent flows. In addition the following studies were performed: a comparison of 4th-, 8th-, 12th- and DRP spatial differencing schemes, the effect of the solution filtering on the results, the effect of large-eddy simulation sub-grid scale models, and the effect of high-order discretization of the viscous terms.

DeBonis, James R.

2013-01-01

391

International Nuclear Information System (INIS)

A wide-angle, split-step finite-difference method with the classical local one-dimensional scheme is presented to analyze the 3-D semi-vectorial wave equation. The method requires only matrix multiplication for beam propagation. To validate the effectiveness, numerical results for the eigen-mode propagation in tilted step-index channel waveguides are studied, and results show that the method has high accuracy and numerical efficiency. (fundamental areas of phenomenology(including applications))

392

Low-truncation-error finite difference equations for photonics simulation 1: Beam propagation

Energy Technology Data Exchange (ETDEWEB)

A methodology is presented that allows the derivation of low-truncation-error finite difference equations for photonics simulation. This methodology is applied to the case of wide-angle beam propagation in two dimensions, resulting in finite difference equations for both TE and TM polarization that are quasi-fourth-order accurate even in the presence of interfaces between dissimilar dielectrics. This accuracy is accomplished without an appreciable increase in numerical overhead and is concretely demonstrated for two test problems having known solutions. These finite difference equations facilitate an approach to the ideal of grid-independent computing and should allow the solution of interesting problems on personal computers.

Hadley, G.R.

1997-08-01

393

Rayleigh Wave Numerical Dispersion in a 3D Finite-Difference Algorithm

A Rayleigh wave propagates laterally without dispersion in the vicinity of the plane stress-free surface of a homogeneous and isotropic elastic halfspace. The phase speed is independent of frequency and depends only on the Poisson ratio of the medium. However, after temporal and spatial discretization, a Rayleigh wave simulated by a 3D staggered-grid finite-difference (FD) seismic wave propagation algorithm suffers from frequency- and direction-dependent numerical dispersion. The magnitude of this dispersion depends critically on FD algorithm implementation details. Nevertheless, proper gridding can control numerical dispersion to within an acceptable level, leading to accurate Rayleigh wave simulations. Many investigators have derived dispersion relations appropriate for body wave propagation by various FD algorithms. However, the situation for surface waves is less well-studied. We have devised a numerical search procedure to estimate Rayleigh phase speed and group speed curves for 3D O(2,2) and O(2,4) staggered-grid FD algorithms. In contrast with the continuous time-space situation (where phase speed is obtained by extracting the appropriate root of the Rayleigh cubic), we cannot develop a closed-form mathematical formula governing the phase speed. Rather, we numerically seek the particular phase speed that leads to a solution of the discrete wave propagation equations, while holding medium properties, frequency, horizontal propagation direction, and gridding intervals fixed. Group speed is then obtained by numerically differentiating the phase speed with respect to frequency. The problem is formulated for an explicit stress-free surface positioned at two different levels within the staggered spatial grid. Additionally, an interesting variant involving zero-valued medium properties above the surface is addressed. We refer to the latter as an implicit free surface. Our preliminary conclusion is that an explicit free surface, implemented with O(4) spatial FD operators and positioned at the level of the compressional stress components, leads to superior numerical dispersion performance. Phase speeds measured from fixed-frequency synthetic seismograms agree very well with the numerical predictions. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Company, for the US Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

Preston, L. A.; Aldridge, D. F.

2010-12-01

394

Unconditionally stable time marching scheme for Reynolds stress models

Progress toward a stable and efficient numerical treatment for the compressible Favre-Reynolds-averaged Navier-Stokes equations with a Reynolds-stress model (RSM) is presented. The mean-flow and the Reynolds stress model equations are discretized using finite differences on a curvilinear coordinates mesh. The convective flux is approximated by a third-order upwind biased MUSCL scheme. The diffusive flux is approximated using second-order central differencing, based on a full-viscous stencil. The novel time-marching approach relies on decoupled, implicit time integration, that is, the five mean-flow equations are solved separately from the seven Reynolds-stress closure equations. The key idea is the use of the unconditionally positive-convergent implicit scheme (UPC), originally developed for two-equation turbulence models. The extension of the UPC scheme for RSM guarantees the positivity of the normal Reynolds-stress components and the turbulence (specific) dissipation rate for any time step. Thanks to the UPC matrix-free structure and the decoupled approach, the resulting computational scheme is very efficient. Special care is dedicated to maintain the implicit operator compact, involving only nearest neighbor grid points, while fully supporting the larger discretized residual stencil. Results obtained from two- and three-dimensional numerical simulations demonstrate the significant progress achieved in this work toward optimally convergent solution of Reynolds stress models. Furthermore, the scheme is shown to be unconditionally stable and positive.

Mor-Yossef, Y.

2014-11-01

395

The Finite Difference Time Domain (FDTD) method is a simple yet powerful method for numerically solving electromagnetic wave phenomenon on computers. The FDTD technique discretizes Maxwell's equations with finite difference equations. These finite difference equations, which approximate local field behavior, are applied to large spatial lattices allowing calculation of a vast array of electromagnetical phenomenon. The greatest strengths of the FDTD method are in its simplicity, efficiency, and diversity. FDTD is capable of modeling the scattering and coupling to lossy dielectrics, lossy magnetics, anisotropic media, dispersive media, and nonlinear materials for general geometric shapes. Wideband frequency information can be obtained using FDTD for both near and far field observation points in a single computational run. However, along with all of its benefits, the FDTD algorithm has some deficiencies. For most problems of interest, poor accuracy at geometry interfaces of differing media and at outer problem space boundarys where the spatial lattice must be truncated are the two largest error sources of the FDTD algorithm. Although most accuracy issues can be circumvented by expending large amounts of computer memory and cpu time, using excessive computer resources is not always possible and is never appealing. The purpose of this thesis is to generalize, analyze, and test various mainstream local Outer Radiating Boundary Conditions (ORBCs) for the FDTD method applied to Maxwell's equations in order to help gain a better understanding of present ORBC limitations. A common mathematical model is presented for the boundary conditions. Boundary conditions shown to fit the model include Mur, Superabsorption, Liao, Higdon, and Lindman ORBCs of varying orders. Simple operators are defined and then used to generate the final discretized equations for each of the boundary conditions, automatically, without requiring complicated high order equations. The procedure also allows the concatenation of different boundary into new higher order boundary conditions broadening the scope of existing ORBCs. Each of the boundary conditions along with various combined boundary conditions are tested in two and three dimensions for a host of typical "real world" scattering and radiation problems. The test results of each boundary condition are then compared to their reflection coefficients for propagating and evanescent plane waves. ORBC stability issues are discussed and several stabilized and unstablized ORBCs are compared and contrasted.

Steich, David James

1995-01-01

396

Energy Technology Data Exchange (ETDEWEB)

An anisotropic constitutive relation was incorporated into the Lagrangian finite-difference wavecode TOODY. The details of the implementation of the constitutive relation in the wavecode and an example of its use are discussed. 4 figures, 1 table.

Swegle, J.W.; Hicks, D.L.

1979-05-01

397

A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.

Fix, G. J.; Rose, M. E.

1983-01-01

398

The accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity is considered. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points.

Bauld, N. R., Jr.; Goree, J. G.

1983-01-01

399

Vibration analysis of rotating turbomachinery blades by an improved finite difference method

The problem of calculating the natural frequencies and mode shapes of rotating blades is solved by an improved finite difference procedure based on second-order central differences. Lead-lag, flapping and coupled bending-torsional vibration cases of untwisted blades are considered. Results obtained by using the present improved theory have been observed to be close lower bound solutions. The convergence has been found to be rapid in comparison with the classical first-order finite difference method. While the computational space and time required by the present approach is observed to be almost the same as that required by the first-order theory for a given mesh size, accuracies of practical interest can be obtained by using the improved finite difference procedure with a relatively smaller matrix size, in contrast to the classical finite difference procedure which requires either a larger matrix or an extrapolation procedure for improvement in accuracy.

Subrahmanyam, K. B.; Kaza, K. R. V.

1985-01-01

400

Enhancing coronary Wave Intensity Analysis robustness by high order central finite differences

Background Coronary Wave Intensity Analysis (cWIA) is a technique capable of separating the effects of proximal arterial haemodynamics from cardiac mechanics. Studies have identified WIA-derived indices that are closely correlated with several disease processes and predictive of functional recovery following myocardial infarction. The cWIA clinical application has, however, been limited by technical challenges including a lack of standardization across different studies and the derived indices' sensitivity to the processing parameters. Specifically, a critical step in WIA is the noise removal for evaluation of derivatives of the acquired signals, typically performed by applying a Savitzky–Golay filter, to reduce the high frequency acquisition noise. Methods The impact of the filter parameter selection on cWIA output, and on the derived clinical metrics (integral areas and peaks of the major waves), is first analysed. The sensitivity analysis is performed either by using the filter as a differentiator to calculate the signals' time derivative or by applying the filter to smooth the ensemble-averaged waveforms. Furthermore, the power-spectrum of the ensemble-averaged waveforms contains little high-frequency components, which motivated us to propose an alternative approach to compute the time derivatives of the acquired waveforms using a central finite difference scheme. Results and Conclusion The cWIA output and consequently the derived clinical metrics are significantly affected by the filter parameters, irrespective of its use as a smoothing filter or a differentiator. The proposed approach is parameter-free and, when applied to the 10 in-vivo human datasets and the 50 in-vivo animal datasets, enhances the cWIA robustness by significantly reducing the outcome variability (by 60%). PMID:25187852

Rivolo, Simone; Asrress, Kaleab N.; Chiribiri, Amedeo; Sammut, Eva; Wesolowski, Roman; Bloch, Lars ?.; Gr?ndal, Anne K.; H?nge, Jesper L.; Kim, Won Y.; Marber, Michael; Redwood, Simon; Nagel, Eike; Smith, Nicolas P.; Lee, Jack

2014-01-01

401

A fourth order finite difference method for solving elliptic partial differential equations

International Nuclear Information System (INIS)

A fourth order finite difference method for solving second order elliptic partial differential equations with Dirichlet boundary conditions on a rectangle is developed. The simplicity of the standard 5-POINT STAR finite difference method (which is second order accurate) in two dimensions is retained. Four problems with known exact solutions are used to test the method and it is demonstrated that, for smooth problems, the method is indeed fourth order accurate. (author)

402

A novel 3-D higher-order finite-difference time-domain framework with complex frequency-shifted perfectly matched layer for the modeling of wave propagation in cold plasma is presented. Second- and fourth-order spatial approximations are used to discretize Maxwell's curl equations and a uniaxial perfectly matched layer with the complex frequency-shifted equations is introduced to terminate the computational domain. A numerical dispersion study of second- and higher-order techniques is elaborated and their stability criteria are extracted for each scheme. Comparisons with analytical solutions verify the accuracy of the proposed methods and the low dispersion error of the higher-order schemes.

Prokopidis, Konstantinos P

2013-01-01

403

A collocated method for the incompressible Navier-Stokes equations inspired by the Box scheme

We present a new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity-pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier-Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.

Ranjan, R.; Pantano, C.

2013-01-01

404

High resolution schemes for hyperbolic conservation laws

International Nuclear Information System (INIS)

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an apppropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes

405

International Nuclear Information System (INIS)

In the lagrangian calculations of some nuclear reactor problems such as a bubble expansion in the core of a fast breeder reactor, the crash of an airplane on the external containment or the perforation of a concrete slab by a rigid missile, the mesh may become highly distorted. A remesh is then necessary to continue the calculation with precision and economy. Similarly, an eulerian calculation of a fluid volume bounded by lagrangian shells can be facilitated by a remesh scheme with continuously adapts the boundary of the eulerian domain to the lagrangian shell. This paper reviews available remesh algorithms for finite element and finite difference calculations of solid and fluid continuum mechanics problems, and presents an improved Finite Element Remesh Method which is independent of the quantities at the nodal points (NP) and the integration points (IP) and permits a restart with a new mesh. (orig.)

406

In the urban environment, surface temperatures and conductive heat fluxes through solid media (roofs, walls, roads and vegetated surfaces) are of paramount importance for the comfort of residents (indoors) and for microclimatic conditions (outdoors). Fully discrete numerical methods are currently used to model heat transfer in these solid media in parametrisations of built surfaces commonly used in weather prediction models. These discrete methods usually use finite difference schemes in both space and time. We propose a spatially-analytical scheme where the temperature field and conductive heat fluxes are solved analytically in space. Spurious numerical oscillations due to temperature discontinuities at the sublayer interfaces can be avoided since the method does not involve spatial discretisation. The proposed method is compared to the fully discrete method for a test case of one-dimensional heat conduction with sinusoidal forcing. Subsequently, the analytical scheme is incorporated into the offline version of the current urban canopy model (UCM) used in the Weather Research and Forecasting model and the new UCM is validated against field measurements using a wireless sensor network and other supporting measurements over a suburban area under real-world conditions. Results of the comparison clearly show the advantage of the proposed scheme over the fully discrete model, particularly for more complicated cases.

Wang, Zhi-Hua; Bou-Zeid, Elie; Smith, James A.

2011-02-01

407

The finite difference time domain (FDTD) algorithm is a popular tool for photonics design and simulations, but it also can yield deep insights into the fundamental nature of light and - more speculatively - into the discretization and connectivity and geometry of space-time. The CFL stability limit in FDTD can be interpreted as a limit on the speed of light. It depends not only on the dimensionality of space-time, but also on its connectivity. Thus the speed of light not only tells us something about the dimensionality of space-time but also about its connectivity. The computational molecule in conventional 2-D FDTD is (? +/- h,y)-(x,+/- y h)-(x-y), where h= triangle x = triangle y . It yields the CFL stability limit ctriangle/hphysically exist). The Green's function of the FDTD model, which differs from that of the wave equation, may tell us something about underlying periodicities in space-time. It may be possible to experimentally observe effects of space-time discretization and connectivity in optics experiments.

Cole, James B.

2014-09-01

408

The validity of the modified equation stability analysis introduced by Warming and Hyett was investigated. It is shown that the procedure used in the derivation of the modified equation is flawed and generally leads to invalid results. Moreover, the interpretation of the modified equation as the exact partial differential equation solved by a finite-difference method generally cannot be justified even if spatial periodicity is assumed. For a two-level scheme, due to a series of mathematical quirks, the connection between the modified equation approach and the von Neuman method established by Warming and Hyett turns out to be correct despite its questionable original derivation. However, this connection is only partially valid for a scheme involving more than two time levels. In the von Neumann analysis, the complex error multiplication factor associated with a wave number generally has (L-1) roots for an L-level scheme. It is shown that the modified equation provides information about only one of these roots.

Chang, Sin-Chung

1987-01-01

409

International Nuclear Information System (INIS)

The effect of magnetic field line ergodization that eliminates magnetic surfaces (either by a resonant magnetic perturbation like in TEXTOR-DED or by intrinsic plasma effects like in W7-X) imposes the need for plasma transport models being able to describe this properly. To handle the ergodicity the concept of local magnetic coordinates allowing a correct discretization of the transport equations with minimized numerical errors is used. For the simulation of plasma transport in perturbed volume, a numerical method based on the finite difference concept has been developed, using a custom-tailored unstructured grid in local magnetic coordinates. This grid is generated by field line tracing to guarantee complete separation of the large parallel transport along B and that perpendicular to B and the ergodicity of the magnetic field does not limit applicability of the method in contrast to the methods based on finite volume ansatz. Perpendicular and parallel fluxes can be effectively separated in our approach and treated independently in the numerical method which has been implemented in the FINDIF code. The finite difference code FINDIF is used to investigate the energy transport in the complex 3D TEXTOR-DED tokamak geometry, where the plasma structures and transport are closely related to the structure of the magnetic field lines. Numerical grids have been prepared in order to simulate 12/4 and 6/2 modes of the DED operation, respectively. In particular, the question, what is the role of long and short magnetic field lines in the heat transfer from the core plasma to divertor surface, is addressed. Simulation results are compared with experimentally determined temperature profiles and heat fluxes at the target

410

A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics

Directory of Open Access Journals (Sweden)

Full Text Available The obesity epidemic is considered a health concern of paramount importance in modern society. In this work, a nonstandard finite difference scheme has been developed with the aim to solve numerically a mathematical model for obesity population dynamics. This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. The construction of the proposed discrete scheme is developed such that it is dynamically consistent with the original differential equations model. Since the total population in this mathematical model is assumed constant, the proposed scheme has been constructed to satisfy the associated conservation law and positivity condition. Numerical comparisons between the competitive nonstandard scheme developed here and Euler's method show the effectiveness of the proposed nonstandard numerical scheme. Numerical examples show that the nonstandard difference scheme methodology is a good option to solve numerically different mathematical models where essential properties of the populations need to be satisfied in order to simulate the real world.

Gilberto González-Parra

2008-12-01

411

Formulation of coarse mesh finite difference to calculate mathematical adjoint flux

International Nuclear Information System (INIS)

The objective of this work is the obtention of the mathematical adjoint flux, having as its support the nodal expansion method (NEM) for coarse mesh problems. Since there are difficulties to evaluate this flux by using NEM. directly, a coarse mesh finite difference program was developed to obtain this adjoint flux. The coarse mesh finite difference formulation (DFMG) adopted uses results of the direct calculation (node average flux and node face averaged currents) obtained by NEM. These quantities (flux and currents) are used to obtain the correction factors which modify the classical finite differences formulation . Since the DFMG formulation is also capable of calculating the direct flux it was also tested to obtain this flux and it was verified that it was able to reproduce with good accuracy both the flux and the currents obtained via NEM. In this way, only matrix transposition is needed to calculate the mathematical adjoint flux. (author)

412

This paper presents an implementation of a 2.5-D finite-difference (FD) code to model acoustic full waveform monopole logging in cylindrical coordinates accelerated by using the new parallel computing devices (PCDs). For that purpose we use the industry open standard Open Computing Language (OpenCL) and an open-source toolkit called PyOpenCL. The advantage of OpenCL over similar languages is that it allows one to program a CPU (central processing unit) a GPU (graphics processing unit), or multiple GPUs and their interaction among them and with the CPU, or host device. We describe the code and give a performance test in terms of speed using six different computing devices under different operating systems. A maximum speedup factor over 34.2, using the GPU is attained when compared with the execution of the same program in parallel using a CPU quad-core. Furthermore, the results obtained with the finite differences are validated using the discrete wavenumber method (DWN) achieving a good agreement. To provide the Geoscience and the Petroleum Science communities with an open tool for numerical simulation of full waveform sonic logs that runs on the PCDs, the full implementation of the 2.5-D finite difference with PyOpenCL is included.

Iturrarán-Viveros, Ursula; Molero, Miguel

2013-07-01

413

A new approach for multisource three-dimensional (3-D) finite-difference (FD) electromagnetic (EM) modeling in the frequency domain is introduced. This approach is based on the balance method, solves for the anomalous electric field and automatically takes into account the conservation law of Maxwell's equations. Also a new Dirichlet boundary condition, based on the quasi-analytical (QA) approximation/series, is proposed to truncate the FD modeling grid significantly without notable loss of computational accuracy. The developed 3-D FD modeling code can be used in different geophysical applications including magnetotelluric (MT), controlled source magnetotelluric (CSMT), airborne, and borehole EM methods in the frequency domain. The modeling results obtained by the new algorithm demonstrated good agreement with the respective integral equation solutions. Regularization methods search for a smooth or focused class of the geoelectrical models to invert for. The traditional approach has been to use smooth models to describe the conductivity distribution in the subsurface formations. A new method for two-dimensional (2-D) MT focusing inversion is developed. It approximates the conductivity distribution by models with blocky (focused) conductivity structures. The class (smooth or focused) of inverse models is chosen based on the objective of the survey and available geological information, and can be determined from inversion by selecting the corresponding stabilizing functional in the regularized objective functional subjected to minimization. This new method was applied to synthetic MT data, and MT field data collected for crustal imaging in Carrizo Plain, California and for mining exploration in Voisey's Bay, Canada. A novel algorithm for 3-D EM iterative migration is introduced. It does not require Frechet (Jacobian) matrix computation but rather the conjugate of Frechet matrix acting on the residual field using just one forward modeling run. This algorithm utilizes the 3-D FD method described above and the regularized conjugate gradient (RCG) scheme. In the framework of this approach, the 3-D FD forward modeling solution is computed three times per frequency in each iteration step. In the first forward solution, the FD forward operator is applied to compute the predicted field for a given conductivity distribution. The residual field is then computed by taking the difference between the observed and predicted data. In the second forward solution, the FD operator is utilized to migrate the residual field in the lower half-space using the adjoint operator. In the third forward solution, the FD operator is used to compute the optimal step of minimization. The practical effectiveness of the newly developed 3-D FD inversion technique is illustrated by inverting both synthetic data and field data of Voisey's Bay like geoelectrical model.

Mehanee, Salah Abdelraheem

414

A Second Order Finite Difference Approximation for the Fractional Diffusion Equation

Directory of Open Access Journals (Sweden)

Full Text Available We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.

H. M. Nasir

2013-07-01

415

Analysis of equilibrium in a tokamak by the finite-difference method

International Nuclear Information System (INIS)

Ideal magnetohydrodynamic equilibrium in a Tokamak having a small radius with an elongated rectangular cross section is studied by applying the finite-difference method to the Grad-Shafranov equation to determine possible limitations for *b=8*pPsup(2)/Bsup(2). The coupled first-order differential equations resulting from the finite-difference Grad-Shafranov equation is solved by the numarical method:1)We concluded that equilibrium consideration alone gives no limitation even for *b approx.1. 2)We have obtained the equilibrium magnetic field configuration charcterized by a set of three parameters;the aspect ratio, *b,and the safety factor. (Author)