Numerical dispersion and dissipation of the triangle-based discontinuous Galerkin method for acoustic and elastic velocity-stress equations

Abstract

In this work, we present the numerical dispersion and dissipation analyses of the triangle-based discontinuous Galerkin method (DGM) for acoustic and elastic velocity-stress equations. The analysis is based on the eigenvalue problem. The applicability of the flux-based DGM is affected by many factors, including numerical flux, basis function, mesh element, and time-stepping scheme. Based on the semi-discrete analysis, we compare different numerical fluxes—Centred flux, local Lax-Friedrichs (LLF) flux, and Godunov flux, different basis functions—Koornwinder-Dubiner (KD) polynomial, Legendre polynomial, and simple monomial function, and four types of triangular meshes. The fully discrete schemes include the third-order total variation diminishing Runge-Kutta (TVD RK) method and the third-order weighted Runge–Kutta (WRK) method. The results indicate that numerical dispersion and dissipation behave differently for different numerical fluxes and mesh types, which verifies the importance of considering numerical fluxes and mesh types. We find that using LLF and Godunov fluxes, and the mesh pattern of case 3 has attractive advantages for numerical simulation. The result of the dispersion-dissipation analysis with different basis functions demonstrates that for high-order integral accuracy, three kinds of basis functions have the same performance in suppressing numerical dispersion and dissipation in triangular elements. The fully discrete dispersion–dissipation analysis shows that the time-stepping scheme introduces numerical dispersion and dissipation and the fully discrete case with a small sampling ratio and Courant number has a better effect in suppressing numerical dispersion and dissipation. Finally, we provide several numerical experiments to validate our theoretical finding, and numerical results are consistent with theoretical dispersion-dissipation analyses.