\(O(\tau^ 2 + h^ 4)\) finite difference scheme for decoupled system of two quasilinear parabolic equations
type: Item
instance of: scholarly article
title: \(O(\tau^ 2 + h^ 4)\) finite difference scheme for decoupled system of two quasilinear parabolic equations
zbMATH Open document ID: 0873.65091
DOI: 10.1016/0377-0427(95)00179-4
author: Tadeusz Styś
published in: Journal of Computational and Applied Mathematics
publication date: 1997-10-30T00:00:00Z
review text: For \(0<x<1\), \(t>0\) the author considers the following system of equations: \[ p_i(t,x) {\partial u_i \over\partial t} (t,x)= a_i(t,x) {\partial^2 u_i \over\partial x^2} (t,x)+ f_i(t,x, u_1(t,x), u_2(t,x)),\quad i=1,2, \] together with initial and homogeneous Dirichlet boundary conditions under appropriate regularity assumptions on \(p_i\), \(a_i\), \(f_i\), \(i=1,2\). Applying a five-points finite difference operator to \({\partial^2u_i \over \partial x^2} (i=1,2)\) in mesh points \(x_j =jh\) \((h>0)\), \(j=1, \dots, N-1\), he obtains the system of ordinary differential equations. He proves that this method of lines is globally convergent having the error \(O(h^4)\). Next, the trapezoidal rule is used for obtaining a fully discrete scheme, which is solved by an implicit iterative process. Numerical results are given for \(f_1= \exp (-u_1 +u_2)\), \(f_2= \exp (u_1-u_2)\).
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 908345
zbMATH Keywords: finite difference method
zbMATH Keywords: method of lines
zbMATH Keywords: local overconvergence
zbMATH Keywords: quasilinear parabolic equations
zbMATH Keywords: error bound
zbMATH Keywords: global convergence
reviewed by: S. Burys
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00179-4
OpenAlex ID: W2135108922
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description: scientific article; zbMATH DE number 908345