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$ L^2$ stable discontinuous Galerkin methods for one-dimensional two-way wave equations

Authors: Yingda Cheng, Ching-Shan Chou, Fengyan Li and Yulong Xing
Journal: Math. Comp. 86 (2017), 121-155
MSC (2010): Primary 35L05, 35L45, 65M12, 65M60
Published electronically: March 3, 2016
MathSciNet review: 3557796
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Abstract: Simulating wave propagation is one of the fundamental problems in scientific computing. In this paper, we consider one-dimensional two-way wave equations, and investigate a family of $ L^2$ stable high order discontinuous Galerkin methods defined through a general form of numerical fluxes. For these $ L^2$ stable methods, we systematically establish stability (hence energy conservation), error estimates (in both $ L^2$ and negative-order norms), and dispersion analysis. One novelty of this work is to identify a sub-family of the numerical fluxes, termed $ \alpha \beta $-fluxes. Discontinuous Galerkin methods with $ \alpha \beta $-fluxes are proven to have optimal $ L^2$ error estimates and superconvergence properties. Moreover, both the upwind and alternating fluxes belong to this sub-family. Dispersion analysis, which examines both the physical and spurious modes, provides insights into the sub-optimal accuracy of the methods using the central flux and the odd degree polynomials, and demonstrates the importance of numerical initialization for the proposed non-dissipative schemes. Numerical examples are presented to illustrate the accuracy and the long-term behavior of the methods under consideration.

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Additional Information

Yingda Cheng
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Ching-Shan Chou
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Fengyan Li
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180

Yulong Xing
Affiliation: Computer Science and Mathematics Division, Oak Ridge Nationalist Laboratory, Oak Ridge, Tennessee 37831 – and – Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Address at time of publication: Department of Mathematics, University of California Riverside, Riverside, California 92521

Received by editor(s): May 6, 2014
Received by editor(s) in revised form: March 28, 2015, and June 26, 2015
Published electronically: March 3, 2016
Additional Notes: The research of the first author was supported by NSF grants DMS-1217563 and DMS-1318186
The research of the second author was supported by NSF grants DMS-1020625 and DMS-1253481
The third author was supported in part by NSF grants DMS-0847241 and DMS-1318409
The research of the fourth author was sponsored by NSF grant DMS-1216454, ORNL and the U. S. Department of Energy, Office of Advanced Scientific Computing Research. The work was partially performed at ORNL, which is managed by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725
Article copyright: © Copyright 2016 American Mathematical Society