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Trigonometric integrators for quasilinear wave equations

Authors: Ludwig Gauckler, Jianfeng Lu, Jeremy L. Marzuola, Frédéric Rousset and Katharina Schratz
Journal: Math. Comp. 88 (2019), 717-749
MSC (2010): Primary 65M15; Secondary 65P10, 65L70, 65M20
Published electronically: May 4, 2018
MathSciNet review: 3882282
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Abstract: Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semidiscretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical Gårding inequality.

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Additional Information

Ludwig Gauckler
Affiliation: Institut für Mathematik, Freie Universität Berlin, Arnimallee 9, D-14195 Berlin, Germany
MR Author ID: 832307

Jianfeng Lu
Affiliation: Departments of Mathematics, Physics, and Chemistry, Duke University, Box 90320, Durham, North Carolina 27708
MR Author ID: 822782
ORCID: 0000-0001-6255-5165

Jeremy L. Marzuola
Affiliation: Department of Mathematics, UNC-Chapel Hill, CB#3250 Phillips Hall, Chapel Hill, North Carolina 27599
MR Author ID: 787291

Frédéric Rousset
Affiliation: Laboratoire de Mathématiques d’Orsay (UMR 8628), Université Paris-Sud, 91405 Orsay Cedex, France; and Institut Universitaire de France

Katharina Schratz
Affiliation: Fakultät für Mathematik, Karlsruhe Institute of Technology, Englerstr. 2, D-76131 Karlsruhe, Germany
MR Author ID: 990639

Keywords: Quasilinear wave equation, trigonometric integrators, exponential integrators, error bounds, loss of derivatives, energy estimates
Received by editor(s): February 9, 2017
Received by editor(s) in revised form: August 25, 2017, and November 3, 2017
Published electronically: May 4, 2018
Additional Notes: This work was supported by Deutsche Forschungsgemeinschaft (DFG) through project GA 2073/2-1 (LG), SFB 1114 (LG) and SFB 1173 (KS) and by National Science Foundation (NSF) under grants DMS-1454939 (JL), DMS-1312874 (JLM) and DMS-1352353 (JLM)
Article copyright: © Copyright 2018 American Mathematical Society