Convergence of numerical algorithms for semilinear hyperbolic systems
type: Item
instance of: scholarly article
title: Convergence of numerical algorithms for semilinear hyperbolic systems
zbMATH Open document ID: 0947.65099
author: Denise Aregba-Driollet
author: Jean-Marc Mercier
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2000-10-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108505
review text: The paper is concerned with blowing up solutions of semilinear hyperbolic systems and their numerical approximation. It is recalled that such systems, of the form \[ \partial_t u + A \partial_x = f(u) \] with \(A\) diagonal(izable) on \(R\) and \(f\) quadratic, admit a unique bounded solution for initial data in \(W^{1,\infty}\), until a (possibly bounded) blow up time \(T^*\). The authors first show an ``abstract'' result. They consider a system of the previous form and its discretization by an explicit finite difference scheme. Under suitable assumptions (called local uniform stability and local uniform consistency), they show that the scheme converges uniformly on \([0,T_0]\) for \(T_0<T^*\) and that the maximal convergence time coincides with \(T^*\). Then they study a semi-implicit scheme and show a similar result. Subsequently, they focus on the approximation of semilinear systems of wave equations. They construct a scheme that satisfies the stability and consistency assumptions. Finally, numerical experiments are presented. They concern the Broadwell system, \(2\times 2\) systems (either conservative or not), and systems of wave equations (with an insight into \(3\)-dimensional problems). They are in good accordance with the theory.
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 1395899
zbMATH Keywords: semilinear hyperbolic systems
zbMATH Keywords: explicit finite difference scheme
zbMATH Keywords: stability
zbMATH Keywords: consistency
zbMATH Keywords: convergence
zbMATH Keywords: systems of wave equations
zbMATH Keywords: numerical experiments
zbMATH Keywords: Broadwell system
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1395899