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

Yuan Xu; Qiang Zhang

doi:10.1007/s42967-024-00370-5

Communications on Applied Mathematics and Computation ›› : 1 -26. 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 Xu ¹
, Qiang Zhang ²^,b

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Abstract

In this paper, we shall carry out the L$^2$-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constant-coefficient 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 ${\mathcal {Q}}^k$-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 ${\mathcal {Q}}^k$-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.

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Yuan Xu, Qiang Zhang. A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods. Communications on Applied Mathematics and Computation 1-26 DOI:10.1007/s42967-024-00370-5