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

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 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.