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A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media

XIJUN HE1,2 ,  XIAORUI YUE2
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1 Department of Mathematics, School of Mathematics and Statistics, Beijing Technology and Business University (BTBU), Beijing 100048, P.R. China,
2 Department of Mathematics, College of Science, Hainan University, Haikou 570228, P.R. China,
JSE 2019, 28(4), 363–390;
Submitted: 9 June 2025 | Revised: 9 June 2025 | Accepted: 9 June 2025 | Published: 9 June 2025
© 2025 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

He, X.J. and Yue, X.R., 2019. A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media. Journal of Seismic Exploration, 28: 363-391. A high-order weighted Runge-Kutta Discontinuous Galerkin Method for solving 2D acoustic and elastic wave equations in isotropic and anisotropic media is proposed in this paper, which is an extension of the existing first-order and second-order methods to higher-order cases. For this method, second-order seismic wave equations are first transformed into a first-order hyperbolic system, then local Lax-Friedrichs (LLF) numerical flux discontinuous Galerkin formulations for spatial discretization are employed, directly leading to a semi discrete ordinary differential equation (ODE) system. For time discretization, an implicit diagonal Runge-Kutta method is introduced. To avoid solving a large-scale system of linear equations, a two-step explicit iterative process is implemented. In addition, a weighting factor is introduced for the iteration to enrich the method. The basis functions we use are 1 ~ 5' order polynomials, leading to 2'°- and 6 order of spatial accuracy. Numerical properties of the high-order weighted Runge-Kutta Discontinuous Galerkin Method are investigated in detail, including numerical error, stability criteria and numerical dispersion, which validate the superiority of the high order method. The proposed method is then applied to several 2D wave propagation problems in isotropic and anisotropic media, including acoustic-elastic interface problems. Results illustrate that this method can effectively suppress numerical dispersion and provide accurate information on the wave field on coarse mesh. We also compare the proposed method with the finite difference method to investigate the computational efficiency.

Keywords
seismic wave equation
high-order
Discontinuous Galerkin Method
DGM
weighted
numerical dispersion
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Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
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