Accuracy and Stability of the Interior-Penalty Discontinuous Galerkin Method for Acoustic and Elastic Wave Propagation
Abstract:Wavefield simulation is an integral part of full-waveform inversion. Therefore, the accuracy and stability of the numerical methods for forward modeling are of paramount importance for the inversion procedure. In this presentation we will focus on the accuracy and stability analysis of a novel method for wave propagation, namely, the Interior-Penalty Discontinuous Galerkin Method (IP-DGM).
IP-DGM is a generalization of the finite-element method in which the basis functions are discontinuous across the element interfaces and continuity is weakly imposed using penalty terms. The advantages of this method for seismic modeling are that it can handle general non-structured finite-element meshes and any type of boundary conditions, which allow the accommodation of topography and heterogeneities in a natural manner. Furthermore, it is also suitable for parallelization in modern high-performance computers and for local time stepping.
We will show here that the stability of this method strongly depends on the polynomial order of the basis functions and on the choice of the penalty. Furthermore, we give necessary and sufficient stability conditions for the acoustic and elastic cases and for basis functions of orders 1 to 10. Based on this analysis we conclude that the stability conditions of this method are more restrictive than those of other high-order methods like the spectral-element method.
We have also analyzed the accuracy of this method based on a grid-dispersion criteria. The accuracy of this method is similar to that of other high-order finite-element methods. The grid-dispersion analysis shows that the accuracy is greatly improved by using high-order basis functions. In particular, using fourth-order basis functions yields minimal dispersion and negligible numerical anisotropy.
We will finally show examples that illustrate the advantages of this method for seismic modeling.