A note on \(L^{\infty }\) bounds and convergence rates of summation-by-parts schemes
type: Item
instance of: scholarly article
title: A note on \(L^{\infty }\) bounds and convergence rates of summation-by-parts schemes
zbMATH Open document ID: 1306.65256
author: M. Svärd
published in: BIT
publication date: 2014-10-31T00:00:00Z
review text: The author proves that summation-by-parts finite difference schemes approximating linear initial boundary value problems are stable in \(L^\infty\). He considers \[ \begin{aligned} u_t &= P(\partial_x) u + F(x,t),\quad 0\leq x \leq L, \quad t\geq 0,\\u(x,0) &= f(x),\\ L_0(\partial_x,u(0,t),g_0(t)) &= 0,\\ L_1(\partial_x,u(L,t),g_1(t)) &= 0,\end{aligned} \] where \(F,f, g_0, g_1\) are smooth and compatible to sufficiently high order. Together with the standard \(L^2\) bound, the \(L^\infty\) bound is sufficient for optimal convergence rates by applying the results of \textit{M. Svärd} and \textit{J. Nordström} [J. Comput. Phys. 218, No. 1, 333-352 (2006; Zbl 1123.65088)].
reviewed by: Dimitris P. Vartziotis
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35L04
zbMATH DE Number: 6363090
zbMATH Keywords: finite differences
zbMATH Keywords: convergence
zbMATH Keywords: stability
zbMATH Keywords: summation-by-parts
zbMATH Keywords: linear initial boundary value problems
MaRDI profile type: Publication
full work available at URL: s10543-014-0471-7
OpenAlex ID: W2088615679
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DOI: 10.1007/S10543-014-0471-7
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description: scientific article; zbMATH DE number 6363090