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Convergence of finite difference methods for the wave equation in two space dimensions


Authors: Siyang Wang, Anna Nissen and Gunilla Kreiss
Journal: Math. Comp. 87 (2018), 2737-2763
MSC (2010): Primary 65M12, 65M06
DOI: https://doi.org/10.1090/mcom/3319
Published electronically: February 2, 2018
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Abstract: 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.


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Additional Information

Siyang Wang
Affiliation: Division of Scientific Computing, Department of Information Technology, Box 337, SE-75105, Uppsala, Sweden
Address at time of publication: Department of Mathematical Sciences, Chalmers University of Technology – and – University of Gothenburg, SE-41296 Gothenburg, Sweden
Email: siyang.wang@chalmers.se

Anna Nissen
Affiliation: Department of Mathematics, University of Bergen, P.O. Box 7803, N-5020, Bergen, Norway
Address at time of publication: Division of Numerical Analysis, Department of Mathematics, KTH, SE-100 44, Stockholm, Sweden
Email: anissen@kth.se

Gunilla Kreiss
Affiliation: Division of Scientific Computing, Department of Information Technology, Box 337, SE-75105, Uppsala, Sweden
Email: gunilla.kreiss@it.uu.se

DOI: https://doi.org/10.1090/mcom/3319
Keywords: Convergence rate, accuracy, two space dimensions, normal mode analysis, finite difference method, second order wave equation
Received by editor(s): October 29, 2016
Received by editor(s) in revised form: May 18, 2017
Published electronically: February 2, 2018
Additional Notes: This work was performed when the second author was at the University of Bergen and was supported by VISTA (project 6357) in Norway. The first author and the third author are partially supported by the Swedish Research Council (project 106500511).
Article copyright: © Copyright 2018 American Mathematical Society

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