Nonstandard finite difference schemes for reaction‐diffusion equations

Abstract

We construct finite difference schemes for a particular class of one-space dimension, nonlinear reactiondiffusion 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

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

“…Our main conclusion is that all the results of the previous article [3] on nonstandard finite difference schemes for reaction-diffusion PDEs can be directly carried over to the case where linear advection is included. The method was applied to the Fisher equation, where it was shown that the scheme not only satisfied the positivity condition, but also upheld the boundedness requirement.…”

An Introduction to Nonstandard Finite Difference Schemes

“…Examples include nonlinear conservative oscillator systems 27 ' 28 and one-dimensional reaction-diffusion equations. 32 An important class of PDEs for which nonstandard techniques have not had much success is that involving Schrodinger type 17,20 or parabolic wave equations. 29 We have made some small progress in this area, 30 ' 31 but to date little of major significance has arisen from these efforts.…”

A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation

“…Mickens [38] considers the general application of nonstandard finite difference schemes to nonlinear reactionadvection equations. Mickens [39] extends these results by considering reaction-diffusion equations and derives positivity preserving nonstandard finite difference schemes. Jordan [40] derives a nonstandard finite difference scheme satisfying a positivity condition to model heat transfer with a quartic nonlinearity.…”