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

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.