Acoustic wave simulation in strongly heterogeneous models using a discontinuous Galerkin method

In recent years, the discontinuous Galerkin method (DGM) has been rapidly developed for the numerical simulation of seismic waves. For wavefield propagation between two adjacent elements, it is common practice to apply a numerical flux to the boundary of each element to propagate waves between adjacent elements. Several fluxes, including the center, penalty, local Lax-Friedrich (LLF), upwind, and Rankine-Hugoniot jump condition-based (RH condition) fluxes, are widely used in numerical seismic wave simulation. However, some fluxes do not account for media differences between adjacent elements. Although different fluxes have been successfully used in DGM for many velocity models, it is unclear whether they can produce sufficiently accurate or stable results for strongly heterogeneous models, such as checkerboard models commonly used in tomographic studies. We test different fluxes using the acoustic wave equation. We analyze the accuracy of the penalty, LLF, upwind, and RH-condition fluxes based on the results of the numerical simulations of the homogeneous and two-layer models. We conduct simulations using checkerboard models, and the results indicate that the LLF, penalty, and upwind fluxes may have instability problems in heterogeneous models with long-time simulations. We observe instability issues in the LLF, penalty, and upwind fluxes when the wave-impedance contrast is high at the media interface. However, the results of the RH-condition flux remain consistently stable. The series of numerical examples presented in this work provide insights into the characteristics and application of fluxes for seismic wave modeling.