Fractional step Runge-Kutta methods for time dependent coefficient parabolic problems
type: Item
instance of: scholarly article
title: Fractional step Runge-Kutta methods for time dependent coefficient parabolic problems
zbMATH Open document ID: 1028.65059
DOI: 10.1016/S0168-9274(02)00191-5
author: B. Bujanda
author: J. C. Jorge
published in: Applied Numerical Mathematics
publication date: 2003-05-22T00:00:00Z
review text: For evolution problems in a Hilbert space, with coercive and maximal operator which is a sum of \(n\) simpler, commuting operators, the authors investigate a class of special implicit Runge-Kutta methods and prove consistency, stability and convergence for both the discretization in time only, and for discretization in time and space. In the later case, the approximation of the \(n\) simpler operators is assumed to preserve properties like coerciveness, commutativity and Lipschitz-continuity. To illustrate the theory, the paper presents also numerical results which, however, are disapointing in the sense that they concern the Peaceman-Rachford alternating direction algorithm which, as a numerical method for two-dimensional parabolic equations was fully investigated about thirty years ago (works by D'yakonov, Samarskij, Yanenko, whereas results not assuming commutativity are also known for more than twenty years). Further, the presented numerical results concern also an own, third-order in time method, but for a parabolic equation with first-order derivatives, and here, to compensate for the first-order upwinding, \(h=\text{const}\cdot\tau^3\) is assumed.
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 1915286
zbMATH Keywords: fractional step methods
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: consistency
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: Hilbert space
zbMATH Keywords: maximal operator
zbMATH Keywords: numerical results
zbMATH Keywords: Peaceman-Rachford alternating
zbMATH Keywords: direction algorithm
MaRDI profile type: Publication
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full work available at URL: s0168-9274(02)00191-5
OpenAlex ID: W2056948330
description: scientific article; zbMATH DE number 1915286