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

• Published in: Mathematics of computation 2017-01, Vol.86 (303), p.121-155
• Main Authors: Cheng, Yingda, Chou, Ching-Shan, Li, Fengyan, Xing, Yulong
• Format: Article
• Language: English
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/mcom/3090

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.