Time Stepping and Linear Stability of Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids

The influence of element shape on the stability of a Runge-Kutta Discontinuous Galerkin method is systematically investigated, in order to improve the time step calculation in practical simulation. The maximum time step for stability is determined by comparing the eigenvalue spectrum of the semi-discrete scalar advection operator to the stability region of the Runge-Kutta integrator. Stability analyses are performed with a broad range of structured periodic triangular grids, all elements of each grid having the same shape, so that each element shape can be associated to a stability bound. Maximum Courant numbers are computed for Carpenter's low-storage (4,5) Runge-Kutta scheme, based on three different measures of the element size. Lower values of the maximum Courant number, to be used in practical simulations, are provided, and the accuracy of the CFL condition is assessed for each element size measure. In order to remedy the relative lack of reliability of CFL conditions, a simplified procedure for stability analysis is presented, that can be used for maximum time step calculation in practical simulations. It is shown in two examples involving respectively an unstructured and a hybrid grid, that it compares favorably to the CFL conditions.