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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.
MSC (2010): Primary 65M12, 65M06
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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  • [1] Saul Abarbanel, Adi Ditkowski, and Bertil Gustafsson, On error bounds of finite difference approximations to partial differential equations--temporal behavior and rate of convergence, J. Sci. Comput. 15 (2000), no. 1, 79-116. MR 1829556,
  • [2] Daniel Appelö and Gunilla Kreiss, Application of a perfectly matched layer to the nonlinear wave equation, Wave Motion 44 (2007), no. 7-8, 531-548. MR 2341028,
  • [3] Mark H. Carpenter, David Gottlieb, and Saul Abarbanel, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys. 111 (1994), no. 2, 220-236. MR 1275021,
  • [4] David C. Del Rey Fernández, Jason E. Hicken, and David W. Zingg, Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. & Fluids 95 (2014), 171-196. MR 3189003,
  • [5] David C. Del Rey Fernández, Pieter D. Boom, and David W. Zingg, A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys. 266 (2014), 214-239. MR 3179765,
  • [6] Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR 1386891
  • [7] Bengt Fornberg, Calculation of weights in finite difference formulas, SIAM Rev. 40 (1998), no. 3, 685-691. MR 1642772,
  • [8] Gregor J. Gassner, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput. 35 (2013), no. 3, A1233-A1253. MR 3048217,
  • [9] Bertil Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput. 29 (1975), 396-406. MR 0386296
  • [10] Bertil Gustafsson, High order difference methods for time dependent PDE, Springer Series in Computational Mathematics, vol. 38, Springer-Verlag, Berlin, 2008. MR 2380849
  • [11] Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger, Time-dependent Problems and Difference Methods, 2nd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2013. MR 3235981
  • [12] Thomas Hagstrom and George Hagstrom, Grid stabilization of high-order one-sided differencing II: Second-order wave equations, J. Comput. Phys. 231 (2012), no. 23, 7907-7931. MR 2972875,
  • [13] Jeremy E. Kozdon and Lucas C. Wilcox, Stable coupling of nonconforming, high-order finite difference methods, SIAM J. Sci. Comput. 38 (2016), no. 2, A923-A952. MR 3474853,
  • [14] R. M. J. Kramer, C. Pantano, and D. I. Pullin, Nondissipative and energy-stable high-order finite-difference interface schemes for 2-D patch-refined grids, J. Comput. Phys. 228 (2009), no. 14, 5280-5297. MR 2537855,
  • [15] Heinz-Otto Kreiss and Joseph Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199-215 (English, with Russian summary). MR 0319382,
  • [16] Heinz-Otto Kreiss, Omar E. Ortiz, and N. Anders Petersson, Initial-boundary value problems for second order systems of partial differential equations, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 3, 559-593. MR 2877365,
  • [17] H. O. KREISS AND G. SCHERER, Finite element and finite difference methods for hyperbolic partial differential equations, Mathematical aspects of finite elements in partial differential equations, Symposium proceedings (1974), pp. 195-212.
  • [18] H. O. KREISS AND G. SCHERER, On the existence of energy estimates for difference approximations for hyperbolic systems, Technical Report, Department of Scientific Computing, Uppsala University, 1977.
  • [19] Heinz-O. Kreiss and Lixin Wu, On the stability definition of difference approximations for the initial-boundary value problem, Appl. Numer. Math. 12 (1993), no. 1-3, 213-227. MR 1227187,
  • [20] K. Mattsson and Mark H. Carpenter, Stable and accurate interpolation operators for high-order multiblock finite difference methods, SIAM J. Sci. Comput. 32 (2010), no. 4, 2298-2320. MR 2678102,
  • [21] Ken Mattsson, Frank Ham, and Gianluca Iaccarino, Stable and accurate wave-propagation in discontinuous media, J. Comput. Phys. 227 (2008), no. 19, 8753-8767. MR 2456091,
  • [22] Ken Mattsson, Frank Ham, and Gianluca Iaccarino, Stable boundary treatment for the wave equation on second-order form, J. Sci. Comput. 41 (2009), no. 3, 366-383. MR 2556470,
  • [23] Ken Mattsson and Jan Nordström, Summation by parts operators for finite difference approximations of second derivatives, J. Comput. Phys. 199 (2004), no. 2, 503-540. MR 2091906,
  • [24] A. Nissen, G. Kreiss, and M. Gerritsen, Stability at nonconforming grid interfaces for a high order discretization of the Schrödinger equation, J. Sci. Comput. 53 (2012), no. 3, 528-551. MR 2996445,
  • [25] A. Nissen, G. Kreiss, and M. Gerritsen, High order stable finite difference methods for the Schrödinger equation, J. Sci. Comput. 55 (2013), no. 1, 173-199. MR 3030708,
  • [26] Anna Nissen, Katharina Kormann, Magnus Grandin, and Kristoffer Virta, Stable difference methods for block-oriented adaptive grids, J. Sci. Comput. 65 (2015), no. 2, 486-511. MR 3411275,
  • [27] Magnus Svärd and Jan Nordström, On the order of accuracy for difference approximations of initial-boundary value problems, J. Comput. Phys. 218 (2006), no. 1, 333-352. MR 2267956,
  • [28] Magnus Svärd and Jan Nordström, Review of summation-by-parts schemes for initial-boundary-value problems, J. Comput. Phys. 268 (2014), 17-38. MR 3192433,
  • [29] Siyang Wang and Gunilla Kreiss, Convergence of summation-by-parts finite difference methods for the wave equation, J. Sci. Comput. 71 (2017), no. 1, 219-245. MR 3620220,
  • [30] Siyang Wang, Kristoffer Virta, and Gunilla Kreiss, High order finite difference methods for the wave equation with non-conforming grid interfaces, J. Sci. Comput. 68 (2016), no. 3, 1002-1028. MR 3530997,

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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

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

Gunilla Kreiss
Affiliation: Division of Scientific Computing, Department of Information Technology, Box 337, SE-75105, Uppsala, Sweden

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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