A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

doi:10.1007/s42967-024-00370-5

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (2): 637-662.doi: 10.1007/s42967-024-00370-5

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A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

Yuan Xu¹, Qiang Zhang²

1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, Jiangsu, China;
2 Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu, China

收稿日期:2023-09-29 修回日期:2023-12-21 接受日期:2024-01-09 出版日期:2025-06-20 发布日期:2025-04-21
通讯作者: Qiang Zhang,qzh@nju.edu.cn;Yuan Xu,yuanxu@njnu.edu.cn E-mail:qzh@nju.edu.cn;yuanxu@njnu.edu.cn
基金资助:
Yuan Xu is supported by the NSFC (Grant No. 12301513), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20230374), and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 23KJB110019). Qiang Zhang is supported by the NSFC (Grant No. 12071214).

A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

Yuan Xu¹, Qiang Zhang²

1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, Jiangsu, China;
2 Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu, China

Received:2023-09-29 Revised:2023-12-21 Accepted:2024-01-09 Online:2025-06-20 Published:2025-04-21
Supported by:
Yuan Xu is supported by the NSFC (Grant No. 12301513), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20230374), and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 23KJB110019). Qiang Zhang is supported by the NSFC (Grant No. 12071214).

摘要/Abstract

摘要： In this paper, we shall carry out the L²-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constantcoefficient hyperbolic equation. The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis. This extension looks easy for most cases; however, there are a few troubles with obtaining good stability results under a standard CFL condition, especially, for those Q^k-elements with lower degree k as stated in the one-dimensional case. To overcome this difficulty, we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose. In addition, an optimal error estimate on Q^k-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.

关键词: Runge-Kutta discontinuous Galerkin (RKDG) method, L²-norm stability analysis, Energy analysis, Two-dimensional hyperbolic equation

Abstract: In this paper, we shall carry out the L²-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constantcoefficient hyperbolic equation. The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis. This extension looks easy for most cases; however, there are a few troubles with obtaining good stability results under a standard CFL condition, especially, for those Q^k-elements with lower degree k as stated in the one-dimensional case. To overcome this difficulty, we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose. In addition, an optimal error estimate on Q^k-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.

Key words: Runge-Kutta discontinuous Galerkin (RKDG) method, L²-norm stability analysis, Energy analysis, Two-dimensional hyperbolic equation

中图分类号:

65M12
65M15

Yuan Xu, Qiang Zhang. A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 637-662.

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A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

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