Dispersion-dissipation analysis of quadrilateral- and triangular-based discontinuous Galerkin methods for the visco-acoustic wave equation

The discontinuous Galerkin method (DGM) has been extensively applied to numerically discretize acoustic and elastic wave equations. However, few studies are focused on viscous media. In this study, we conduct a comprehensive numerical dispersion-dissipation analysis of DGM for the visco-acoustic wave equation and simulate wave propagation in viscous media. The plane-wave analysis is based on the standard-linear-solid (SLS) -based model. A weighted Runge- Kutta (WRK) time scheme, Legendre polynomials, and a local Lax-Friedrichs flux are employed. The fully discrete analyses are implemented in quadrilateral and triangular elements. We consider two types of triangular elements, the quality factors and the number of mechanisms. Our results show that the quality factor leads to the numerical dispersion, the value of which is constant within a range of small sampling rate, but does not cause numerical dissipation. The visco-acoustic wave equation for different mechanisms for viscosity also introduces an inherent numerical-dispersion value that is not affected by the sampling rate. Meanwhile, we find that the stability of the quadrilateral mesh is stronger than that of the triangular mesh under the same order of basis functions. Several numerical experiments are provided to validate some theoretical findings. Seismic waves go through attenuation and phase distortion during propagation, and the numerical results indicate that the DGM is suitable for simulating seismic wave propagation in the SLS-based model.