Analysis of a nonlinear difference scheme in reaction-diffusion
type: Item
instance of: scholarly article
title: Analysis of a nonlinear difference scheme in reaction-diffusion
zbMATH Open document ID: 0615.65097
DOI: 10.1007/BF01389703
author: A. R. Mitchell
author: Ben-Yu Guo
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133127
review text: The asymptotic stability and the convergence of a difference scheme for solving the following nonlinear partial differential equation: \[ (1)\quad \partial u/\partial t-\epsilon (\partial^ 2U/\partial x^ 2)- U(1-U)=0;\quad U(0,t)=U(1,t)=0,\quad t\geq 0;\quad U(x,0)=U_ 0(x),\quad 0<x<1; \] is investigated. A rectangular grid with the mesh sizes h and \(\tau\) is considered and the Crank-Nicolson scheme is applied to a linear model of the problem described by (1). It is found that for \(\epsilon <\epsilon^*=h^ 2/4 \sin^ 2(\pi h/2)\) there are solutions with arbitrary small initial values, which can grow without bound for all \(x=ih\). Essentially the same condition \((\epsilon <\epsilon^*)\) is obtained when analyzing the solution of the corresponding steady problem \(\epsilon (\partial^ 2V/\partial x^ 2)+V(1-V)=0\), \(V(0)=V(1)=0\) and demanding that \(u^{(k)}(x)\to v(x)\) as \(k\to \infty\), although the proofs here are much more complicated. Finally, uniform convergence of the scheme is proven and the analysis is compared with numerical results. Almost all ''numerical findings are corroborated by the theoretical results of the paper''.
reviewed by: Jiří Gregor
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 3997076
zbMATH Keywords: parabolic reaction diffusion equation
zbMATH Keywords: spruce budworm problem
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Crank-Nicolson scheme
zbMATH Keywords: uniform convergence
MaRDI profile type: Publication
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OpenAlex ID: W1984791044
description: scientific article; zbMATH DE number 3997076