Modified arc-length adaptive algorithms for degenerate reaction-diffusion equations.
type: Item
instance of: scholarly article
title: Modified arc-length adaptive algorithms for degenerate reaction-diffusion equations.
zbMATH Open document ID: 1036.65072
DOI: 10.1016/S0096-3003(00)00158-2
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: The authors consider the following reaction-diffusion equation \[ u_t= \delta^2 u_{xx}+ p(x) u_x+ f(u),\quad 0< x< 1,\quad t> t_0 \] under homogeneous initial and Dirichlet boundary conditions, where \(0<\delta\leq 1\), \(0\leq p(x)\leq b\), \(0\leq x\leq 1\), \(t\geq t_0\). The singular source term \(f(u)\) is monotonically increasing for \(0\leq u< 1\) and \(f(0)= 1\), \(\lim_{u\to r}\, f(u)= \infty\). Modified monitor functions based on arc-lengths of the rate function \(u_t\) are constructed and used to obtain an adaptive semi-discretization on a non-uniform space grid and a two-stage fully adaptive temporal scheme. It is shown, under some assumptions on the step-size, that the numerical solution well imitates monotonic properties of the physical solution. A numerical example for \(p\equiv 0\), \(f(u)= (1- u)^{-\theta}\), \(\theta> 0\) is investigated in details.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M50
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1861023
zbMATH Keywords: reaction-diffusion equations
zbMATH Keywords: degeneration
zbMATH Keywords: finite difference
zbMATH Keywords: arc-length adaptive algorithms
zbMATH Keywords: semi-discretization
zbMATH Keywords: numerical example
author: Qin Sheng
author: A. Q. M. Khaliq
reviewed by: S. Burys
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1861023