A new positivity‐preserving nonstandard finite difference scheme for the DWE

An improved positivity‐preserving nonstandard finite difference scheme for the linear damped wave equation is presented. Unlike an earlier such scheme developed by the authors, the new scheme involves three time levels and is therefore able to include the effects of the equation's relaxation coefficient. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential, 2005     

A positivity‐preserving nonstandard finite difference scheme for the damped wave equation

A positivity‐preserving nonstandard finite difference scheme is constructed to solve an initial‐boundary value problem involving heat transfer described by the Maxwell‐Cattaneo thermal conduction law, i.e., a modified form of the classical Fourier flux relation. The resulting heat transport equation is the damped wave equation, a PDE of hyperbolic type. In addition, exact analytical solutions are given, special cases are mentioned, and it is noted that the positivity condition is equivalent to the usual linear stability criteria. Finally, solution profiles are plotted and possible extensions to a delayed diffusion equation and nonlinear reaction‐diffusion systems are discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition

An essential feature of nonstandard finite difference schemes for differential equations is the precise manner in which the discretization of derivatives is made. We demonstrate, for differential equations modeling systems where the solutions satisfy a positivity condition, that procedures can be formulated to calculate the so‐called denominator functions that appear in the discrete derivatives. These procedures are applied to a number of both ordinary and partial model differential equations to illustrate their use. © 2006 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.     

A finite difference scheme for traveling wave solutions to Burgers equation

We construct a finite difference scheme for the ordinary differential equation describing the traveling wave solutions to the Burgers equation. This difference equation has the property that its solution can be calculated. Our procedure for determining this solution follows closely the analysis used to obtain the traveling wave solutions to the original ordinary differential equation. The finite difference scheme follows directly from application of the nonstandard rules proposed by Mickens. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 815–820, 1998     

Analysis of a mathematical model for cancer treatment by the nonstandard finite difference methods

We construct two nonstandard finite difference schemes and use them to study a mathematical model of cancer therapy. Several recent studies show various aspects of the immune response against the cancer. Our discrete models emphasize the role of antibodies in any form of therapy by taking into account the development of anticancer therapies (chemotherapy, immunotherapy, radiation therapy). The nonstandard finite difference models are implemented by using Matlab. Numerical simulations show the existence of a separation line between the basins of attraction of cancerous cell-free and the highest equilibrium cancerous cell.     

Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection‐reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210‐221, 2014     

Nonstandard finite difference schemes for reaction‐diffusion equations

We construct finite difference schemes for a particular class of one‐space dimension, nonlinear reaction‐diffusion PDEs. The use of nonstandard finite difference methods and the imposition of a positivity condition constrain the schemes to be explicit and allow the determination of functional relations between the space and time step‐sizes. The general procedure is illustrated by applying it to several important model systems of PDEs © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 201–214, 1999     

Exact finite difference scheme for linear differential equation with constant coefficients

An exact finite difference equation for the n-th order linear differential equation with real, constant coefficients is constructed. The exact finite difference scheme is expressed differently but equivalent to that given by Potts [3].     

Construction of nonstandard finite difference schemes for $1{1\over 2}$ space‐dimension‐coupled PDEs

Two coupled PDEs, where one has a diffusion term and the other does not, are defined to be space‐dimension systems. We show how to construct nonstandard finite difference schemes for such systems and demonstrate that they are positivity‐preserving. These methods also allow the calculation of an explicit functional relationships between the time and space step‐sizes. The case of water flowing through fractured bedrock is used to illustrate our procedure. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Contributions to the mathematics of the nonstandard finite difference method and applications

We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 518–543, 2001     

A finite difference scheme for solving the heat transport equation at the microscale

Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank‐Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999     

A nonstandard finite difference method for solving a mathematical model of HIV-TB co-infection

We design and analyze an unconditionally convergent nonstandard finite-difference method to study transmission dynamics of a mathematical model of HIV-TB co-infection. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves positivity of the solution which is one of the essential requirements when modelling epidemic diseases. Furthermore, we show that the numerical method is unconditionally stable. Competitive numerical results confirming theoretical investigations are provided. Comparisons are also made with other conventional approaches that are routinely used to solve these types of problems.     

Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model

This paper studies the stability properties of a nonstandard finite difference (NSFD) scheme used to simulate the dynamics of a mouse population model in hantavirus epidemics. It is shown that this difference scheme and the underlying system of differential equations have the same dynamics. The proof uses the fact that the total population obeys the logistic equation, as well as techniques from calculus, graphical analysis, and dynamical systems.     

Nonstandard finite difference schemes for reaction‐‐diffusion equations having linear advection

We extend previous work on nonstandard finite difference schemes for one‐space dimension, nonlinear reaction–diffusion PDEs to the case where linear advection is included. The use of a positivity condition allows the determination of a functional relation between the time and space step‐sizes, and provides schemes that are explicit. The Fisher equation is used to illustrate the method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 361–364, 2000     

A dynamically optimized finite difference scheme for Large-Eddy Simulation

A low-dispersive dynamic finite difference scheme for Large-Eddy Simulation is developed. The dynamic scheme is constructed by combining Taylor series expansions on two different grid resolutions. The scheme is optimized dynamically through the real-time adaption of a dynamic coefficient according to the spectral content of the flow, such that the global dispersion error is minimal. In the case of DNS-resolution, the dynamic scheme reduces to the standard Taylor-based finite difference scheme with formal asymptotic order of accuracy. When going to LES-resolution, the dynamic scheme seamlessly adapts to a dispersion-relation preserving scheme. The scheme is tested for Large-Eddy Simulation of Burgers equation. Very good results are obtained.     

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.