Analysis of a nonlinear difference scheme in reaction-diffusion
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''.
scientific article; zbMATH DE number 3997076