Analysis of stability and convergence of finite-difference methods for a reaction-diffusion problem on a one-dimensional growing domain

Mackenzie, J.A. and Madzvamuse, A. (2011) Analysis of stability and convergence of finite-difference methods for a reaction-diffusion problem on a one-dimensional growing domain. IMA Journal of Numerical Analysis, 31 (1). pp. 212-232. ISSN 0272-4979 (https://doi.org/10.1093/imanum/drp030)

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Abstract

In this paper we consider the stability and convergence of finite difference discretisations of a reaction-diffusion equation on a one-dimensional domain which is growing in time. We consider discretisations of conservative and non-conservative formulations of the governing equation and highlight the different stability characteristics of each. Although non-conservative formulations are the most popular to date, we find that discretisations of the conservative formulation inherit greater stability properties. Furthermore, we present a novel adaptive time integration scheme based on the well-known -method and describe how the parameter should be chosen to ensure unconditional stability, independently of the rate of domain growth. This work is a preliminary step towards an analysis of numerical schemes for the solution of reaction-diffusion systems on growing domains. Such problems arise in many practical areas including biological pattern formation and tumour growth.

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