A partial upwind difference scheme for nonlinear parabolic equations
type: Item
instance of: scholarly article
title: A partial upwind difference scheme for nonlinear parabolic equations
zbMATH Open document ID: 0676.65094
DOI: 10.1016/0377-0427(89)90295-1
author: M. C. Pandian
published in: Journal of Computational and Applied Mathematics
publication date: 1989-01-01T00:00:00Z
review text: In many diffusion processes arising in physical problems, essentially the transport term plays the dominant role. In these cases, it is known that the centered differencing of the first order term in space variable in the governing nonlinear parabolic transport equation gives rise to oscillatory solutions at reasonable grid sizes. One-sided or ``upwind'' schemes can give oscillation free solutions. However, in general, full ``upwind'' schemes introduce large amounts of artificial diffusions in contrast to the centered schemes. This paper presents a weighted difference scheme which is a generalization of the scheme due to \textit{S. Osher} [SIAM J. Numer. Anal. 18, 129-144 (1981; Zbl 0471.65069)]. Here two parameters \(a_ 1\), \(a_ 2\) in the difference approximation of the nonlinear first order term are used. \((a_ i=1\) leads to Osher's full upwind form, and \(a_ i=0\) corresponds to the centered form.) Using the method of lower and upper solutions and the theory of M-matrices, the scheme is found to be first- order accurate in the space variable. If Osher's full upwind scheme is used, only first-order approximation is found, too. But the actual errors are smaller for partial upwind difference schemes. To support this, a numerical example for a Burger's equation is given.
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 4108083
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: M-matrix
zbMATH Keywords: difference approximation
zbMATH Keywords: partial upwind factors
zbMATH Keywords: centered form
zbMATH Keywords: upwind form
zbMATH Keywords: explicit method
zbMATH Keywords: error bound
zbMATH Keywords: upwind schemes
zbMATH Keywords: diffusion processes
zbMATH Keywords: nonlinear parabolic transport equation
zbMATH Keywords: oscillatory solutions
zbMATH Keywords: weighted difference scheme
zbMATH Keywords: numerical example
zbMATH Keywords: Burger's equation
MaRDI profile type: Publication
full work available at URL: 0377-0427(89)90295-1
OpenAlex ID: W2081646018
cites work: Monotone Difference Schemes for Singular Perturbation Problems
cites work: Q5558541
cites work: Q3848301
cites work: Q5585767
cites work: Nonlinear Singular Perturbation Problems and One Sided Difference Schemes
cites work: A weighted difference scheme and monotone iterative methods for quasilinear boundary value problems
cites work: Q4113353
cites work: Q5533735
cites work: Operator inequalities
cites work: An analysis of the Petrov - Galerkin finite element method
cites work: On a conservation upwind finite element scheme for convective diffusion equations
cites work: A Numerical Conception of Entropy for Quasi-Linear Equations
description: scientific article; zbMATH DE number 4108083