Symplectic interior penalty discontinuous Galerkin method for solving the seismic scalar wave equation

Abstract

To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.

A modified symplectic discontinuous Galerkin method for acoustic and elastic wave simulations

“…Incorporated with the interior penalty DG method in Section 2.1, we call this method the symplectic DG (SDG) method. He et al (2019) studied this method in great detail.…”

“…To discuss the numerical properties including the stability condition, dispersion and dissipation, plane wave analysis should be carried out first. Many studies have focused on the plane wave analysis of the interior penalty DG method for acoustic and elastic cases (e.g., Ainsworth et al, 2006;de Basabe and Sen, 2010;Ferroni et al, 2017;He et al, 2019;He et al, 2020a). One can follow the steps in the above listed studies to analyze the numerical properties of MSDG method.…”

“…It is precisely because of these advantages that much research work has been carried out on the DG methods (e.g., Cockburn and Shu, 2001;Li, 2006). Among them, there are also many studies on solving wave propagation problems (e.g., Käser and Dumbser, 2006;Rivière, 2008;Etienne et al, 2010;Wilcox et al, 2010;Yang et al, 2016;Ferroni et al, 2017;Meng and Fu, 2018;He et al, 2019;He et al, 2020aHe et al, , 2020bHe X. J. et al, 2022, He et al, 2022.…”

“…Many symplectic time-stepping methods have emerged. (Hairer et al, 2006;Chen, 2009;Iwatsu, 2009;Feng and Qin, 2010;Ma et al, 2017;He et al, 2019). Particularly, Ma and Yang (2017) investigated the symplectic schemes, and suggested a series of modified symplectic partitioned Runge-Kutta methods, and their results suggest that these modified schemes have minimized phase errors.…”

“…This research is an advance based on the work of He et al (2019), in which they used a classic symplectic partitioned Runge-Kutta method proposed by Iwatsu (2009), and they only considered the acoustic wave equation. In this study, a modified symplectic discontinuous Galerkin (MSDG) method is proposed for seismic wavefield modeling, which combines the interior penalty DG method with a modified third-order partitioned Runge-Kutta method proposed by Ma and Yang (2017).…”