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

In this paper we consider the stability and convergence of finite-difference discretizations of a reaction-diffusion equation on a one-dimensional domain that is growing in time. We consider discretizations of conservative and nonconservative formulations of the governing equation and highlight the different stability characteristics of each. Although nonconservative formulations are the most popular to date, we find that discretizations of the conservative formulation inherit greater stability properties. Furthermore, we present a novel adaptive time integration scheme based on the well-known theta method and describe how the parameter theta 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.