A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A high‐order compact boundary value method for solving one‐dimensional heat equations

We combine fourth‐order boundary value methods (BVMs) for discretizing the temporal variable with fourth‐order compact difference scheme for discretizing the spatial variable to solve one‐dimensional heat equations. This class of new compact difference schemes achieve fourth‐order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second‐order Crank‐Nicolson scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 846–857, 2003.     

Time accurate fast wavelet‐Taylor Galerkin method for partial differential equations

We introduce the concept of fast wavelet‐Taylor Galerkin methods for the numerical solution of partial differential equations. In wavelet‐Taylor Galerkin method discretization in time is performed before the wavelet based spatial approximation by introducing accurate generalizations of the standard Euler, θ and leap‐frog time‐stepping scheme with the help of Taylor series expansions in the time step. We will present two different time‐accurate wavelet schemes to solve the PDEs. First, numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for in localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions, and nonlinear equation with a shock in solution in two dimensions. Second, our schemes deal with more regular class of problems where wavelets are not efficient procedure for data compression but we can use the good approximation properties of wavelet. Here time‐accurate schemes lead to consistent mass matrix in an explicit time stepping, which can be solved by approximate factorization techniques. Numerical experiment deals with more regular class of problems like heat equation as well as coupled linear system in two dimensions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

An efficient splitting technique for two‐layer shallow‐water model

We consider the numerical approximation of the weak solutions of the two‐layer shallow‐water equations. The model under consideration is made of two usual one‐layer shallow‐water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one‐layer shallow‐water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow‐water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non‐negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1396–1423, 2015     

A numerical method for the one‐dimensional sine‐Gordon equation

A numerical method based on a predictor–corrector (P‐C) scheme arising from the use of rational approximants of order 3 to the matrix‐exponential term in a three‐time level recurrence relation is applied successfully to the one‐dimensional sine‐Gordon equation, already known from the bibliography. In this P‐C scheme a modification in the corrector (MPC) has been proposed according to which the already evaluated corrected values are considered. The method, which uses as predictor an explicit finite‐difference scheme arising from the second order rational approximant and as corrector an implicit one, has been tested numerically on the single and the soliton doublets. Both the predictor and the corrector schemes are analyzed for local truncation error and stability. From the investigation of the numerical results and the comparison of them with other ones known from the bibliography it has been derived that the proposed P‐C/MPC schemes at least coincide in terms of accuracy with them. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Efficient numerical schemes for solving the self‐consistent field equations of flexible–semiflexible diblock copolymers

We present two efficient iterative schemes for solving the self‐consistent field equations of flexible–semiflexible diblock copolymers. One is a semi‐implicit scheme developed by employing asymptotic expansion, and the other is a hybrid scheme combining the robustness of the steepest descent method with the efficiency of the conjugate gradient method. In our position‐one‐dimensional and position‐two‐dimensional numerical experiments, we demonstrate that these schemes are much more efficient than the steepest descent method. Copyright © 2013 John Wiley & Sons, Ltd.     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

A predictor‐‐corrector scheme for the sine‐Gordon equation

A predictor–corrector scheme is developed for the numerical solution of the sine‐Gordon equation using the method of lines approach. The solution of the approximating differential system satisfies a recurrence relation, which involves the cosine function. Pade' approximants are used to replace the cosine function in the recurrence relation. The resulting schemes are analyzed for order, stability, and convergence. Numerical results demonstrate the efficiency and accuracy of the predictor–corrector scheme over some well‐known existing methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 133–146, 2000     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

Asymptotic‐preserving Godunov‐type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation terms

We devise a new class of asymptotic‐preserving Godunov‐type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation source terms governed by a relaxation time ε. As an alternative to classical operator‐splitting techniques, the objectives of these schemes are twofold: first, to give accurate numerical solutions for large, small, and in‐between values of ε and second, to make optional the choice of the numerical scheme in the asymptotic regime ε tends to zero. The latter property may be of particular interest to make easier and more efficient the coupling at a fixed spatial interface of two models involving very different values of ε. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

Practical finite‐analytic method for solving differential equations by compact numerical schemes

Methodology for development of compact numerical schemes by the practical finite‐analytic method (PFAM) is presented for spatial and/or temporal solution of differential equations. The advantage and accuracy of this approach over the conventional numerical methods are demonstrated. In contrast to the tedious discretization schemes resulting from the original finite‐analytic solution methods, such as based on the separation of variables and Laplace transformation, the practical finite‐analytical method is proven to yield simple and convenient discretization schemes. This is accomplished by a special universal determinant construction procedure using the general multi‐variate power series solutions obtained directly from differential equations. This method allows for direct incorporation of the boundary conditions into the numerical discretization scheme in a consistent manner without requiring the use of artificial fixing methods and fictitious points, and yields effective numerical schemes which are operationally similar to the finite‐difference schemes. Consequently, the methods developed for numerical solution of the algebraic equations resulting from the finite‐difference schemes can be readily facilitated. Several applications are presented demonstrating the effect of the computational molecule, grid spacing, and boundary condition treatment on the numerical accuracy. The quality of the numerical solutions generated by the PFAM is shown to approach to the exact analytical solution at optimum grid spacing. It is concluded that the PFAM offers great potential for development of robust numerical schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, 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, second‐order and 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 nonsingular. 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. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

Investigations of nonstandard,Mickens‐type,finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

It is well known that standard finite‐difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (??(1/r) or ??(log r)) behavior of the solution near the origin (r = 0). New nonstandard finite‐difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these “Mickens‐type” finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co‐ordinates are investigated. The numerical results generated by the Mickens‐type schemes are compared favorably with solutions obtained from standard finite‐difference schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 380–398, 2003.     

High‐resolution monotone schemes based on quasi‐characteristics technique

In this article, we consider a new technique that allows us to overcome the well‐known restriction of Godunov's theorem. According to Godunov's theorem, a second‐order explicit monotone scheme does not exist. The techniques in the construction of high‐resolution schemes with monotone properties near the discontinuities of the solution lie in choosing of one of two high‐resolution numerical solutions computed on different stencils. The criterion for choosing the final solution is proposed. Results of numerical tests that compare with the exact solution and with the numerical solution obtained by the first‐order monotone scheme are presented. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 262–276, 2001     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.     

A note on second‐order finite‐difference schemes on uniform meshes for advection‐‐diffusion equations

An artificial‐viscosity finite‐difference scheme is introduced for stabilizing the solutions of advection‐diffusion equations. Although only the linear one‐dimensional case is discussed, the method is easily susceptible to generalization. Some theory and comparisons with other well‐known schemes are carried out. The aim is, however, to explain the construction of the method, rather than considering sophisticated applications. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 581–588, 1999     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.