CFL Conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We study time step restrictions due to linear stability constraints of Runge–Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax–Friedrichs flux or the upwind flux, and integrated in time with various Runge–Kutta schemes designed for linear wave propagation problems or non-linear applications. Von–Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order p ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax–Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.