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Convergence Of Finite Difference Methods For The Wave Equation in Two Space Dimensions
Dept Informat Technol, Div Sci Comp, Box 337, SE-75105 Uppsala, Sweden.;Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden.;Univ Gothenburg, SE-41296 Gothenburg, Sweden..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Univ Bergen, Dept Math, POB 7803, N-5020 Bergen, Norway.
Dept Informat Technol, Div Sci Comp, Box 337, SE-75105 Uppsala, Sweden..
2018 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 314, p. 2737-2763Article in journal (Refereed) Published
Abstract [en]

When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2018. Vol. 87, no 314, p. 2737-2763
Keywords [en]
Convergence rate, accuracy, two space dimensions, normal mode analysis, finite difference method, second order wave equation
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-233404DOI: 10.1090/mcom/3319ISI: 000440340300006Scopus ID: 2-s2.0-85051019859OAI: oai:DiVA.org:kth-233404DiVA, id: diva2:1240471
Note

QC 20180821

Available from: 2018-08-21 Created: 2018-08-21 Last updated: 2018-10-16Bibliographically approved

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