Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

This paper studies the Neimark–Sacker bifurcation of a diffusive food‐limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank‐Nicolson difference scheme, and nonstandard finite‐difference scheme. The existence of Neimark‐Sacker bifurcation at the equilibrium is obtained. Our results show that Crank‐Nicolson and nonstandard finite‐difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel variable‐order fractional nonlinear Klein Gordon model: A numerical approach

In this article, a novel variable order fractional nonlinear Klein Gordon model is presented where the variable‐order fractional derivative is defined in the Caputo sense. The merit of nonstandard numerical techniques is extended here and we present the weighted average nonstandard finite difference method to study numerically the proposed model. Special attention is paid to study the convergence and to the stability analysis of the numerical technique. Moreover, the truncation error is analyzed. Three test examples are provided. Comparative studies are done between the used numerical technique and the weighted average finite difference method. It is found that the stability regions are larger by using the weighted average nonstandard finite difference method.     

Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation

A competitive nonstandard semi‐explicit finite‐difference method is constructed and used to obtain numerical solutions of the diffusion‐free generalized Nagumo equation. Qualitative stability analysis and numerical simulations show that this scheme is more robust in comparison to some standard explicit methods such as forward Euler and the fourth‐order Runge‐Kutta method (RK4). The nonstandard scheme is extended to construct a semi‐explicit and an implicit scheme to solve the full Nagumo reaction‐diffusion equation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 363–379, 2003.     

An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers

This paper consists in studying a mathematical model of solvent diffusion through the glassy polymers as a one-dimensional moving boundary problem with kinetic undercooling. We establish an iterative variable time-step method based on a nonstandard finite difference (NSFD) scheme to solve the considered moving boundary problem. The monotonicity and positivity of the numerical solution are proved. The numerical approach is investigated for three test problems composed of constant and inconstant diffusion coefficients for different values of parameters to demonstrate the validity and ability of the method.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

A space‐time mixed‐hybrid finite element method for the damped wave equation

A space‐time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed‐hybrid finite element methods, and where an original conforming approximation of H(div;Ω) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations

In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

A discrete duality finite volume discretization of the vorticity‐velocity‐pressure stokes problem on almost arbitrary two‐dimensional grids

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity‐velocity‐pressure formulation of the two‐dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015     

Nonstandard methods for the convective‐dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian‐Lagrangian method is constructed for the one‐dimensional, transient convective‐dispersive transport equation with nonlinear reaction terms. An “exact” difference scheme is applied to the convection‐reaction part of the equation to produce a semi‐discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection‐dominated, reactive transport problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617–624, 1999     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

An energy‐preserving scheme is proposed for the three‐coupled nonlinear Schrödinger (T‐CNLS) equation. The T‐CNLS equation is rewritten into the classical Hamiltonian form. Then the spatial variable is discretized by using high‐order compact method to convert it into a finite‐dimensional Hamiltonian system. Next, a second‐order averaged vector field (AVF) method is employed in time which results in an energy‐preserving scheme. Some theoretical results such as convergence are investigated. In addition, it provides some numerical examples to illustrate the robustness and reliability of the theoretical results. It also explores the role of the parameters in the model and initial condition on the wave propagation.     

BVPs on infinite intervals: A test problem,a nonstandard finite difference scheme and a posteriori error estimator

As far as the numerical solution of boundary value problems defined on an infinite interval is concerned, in this paper, we present a test problem for which the exact solution is known. Then we study an a posteriori estimator for the global error of a nonstandard finite difference scheme previously introduced by the authors. In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from 2 nested quasi‐uniform grids. We observe that if the grids are sufficiently fine, the Richardson error estimate gives an upper bound of the global error.     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition

In this work the combined finite difference and spectral methods have been proposed for the numerical solution of the one‐dimensional wave equation with an integral condition. The time variable is approximated using a finite difference scheme. But the spectral method is employed for discretizing the space variable. The main idea behind this approach is that we can get high‐order results. The new method is used for two test problems and the numerical results are obtained to support our theoretical expectations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability preserving NSFD scheme for a multi‐group SVIR epidemic model

A discrete multi‐group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.