Mathematical Analysis of a Nonstandard Finite Difference Scheme for the Fisher Equation with Nonlinear Diffusion

Article type: Research Article

Authors: Mickens, Ronald E.

Affiliations: Clark Atlanta University, Department of Physics, Atlanta, GA 30314, USA

Abstract: A finite difference scheme is constructed for the Fisher partial differential equation having a nonlinear diffusion term. The application of nonstandard methods and the requirement of a positivity condition leads to a discrete model having many of the dynamical properties of the original differential equation. In particular, the fixed-points have the same linear stability properties as those in the differential equation. Further, the scheme can be rewritten in an explicit form with a definite relation existing between the time and space step-sizes.

Keywords: Nonstandard finite difference schemes, Fisher equation, nonlinear diffusion

Keywords: 65C20, 65D30, 65M06

DOI: 10.3233/JCM-2001-1106

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 1, no. 1, pp. 135-142, 2001