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

doi:10.1007/s42967-024-00370-5

Abstract

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

CLC Number:

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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References

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