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

J.A. Mackenzie, A. Madzvamuse

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10 Citations (Scopus)

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.
LanguageEnglish
Pages212-232
Number of pages21
JournalIMA Journal of Numerical Analysis
Volume31
Issue number1
Early online date9 Nov 2009
DOIs
Publication statusPublished - 2011

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Reaction-diffusion Problems
Convergence of numerical methods
Stability and Convergence
Finite difference method
Difference Method
Finite Difference
Discretization
Formulation
Domain Growth
Unconditional Stability
Tumor Growth
Pattern Formation
Time Integration
Reaction-diffusion System
Reaction-diffusion Equations
Numerical Scheme
Governing equation
Tumors

Keywords

  • reaction-diffusion
  • growing domain
  • stability' bilogical pattern formation
  • exponential growth functions

Cite this

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AB - 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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