Relaxing the CFL Number of the Discontinuous Galerkin Method

We propose a family of high order methods for solution of hyperbolic conservation laws which are based on the discontinuous Galerkin (DG) spatial discretization. In the standard DG method, the dispersion and dissipation errors and the spectrum of the semi-discrete scheme are related to the [ p p+1 ] Padé approximants of exp(z) and exp(−z). These Padé approximants are responsible for the superconvergent O(h2p+2) and O(h2p+1) errors in dispersion and dissipation, respectively, and the restriction of the CFL number when increasing the order of approximation, p. By modifying the DGM we obtain different rational approximations of the exponential, thereby sacrificing some of the superconvergence of the method, and construct new schemes which allow larger time steps than the original DGM while having the same order of convergence in the L2 norm. This is achieved through modifications to the numerical flux. The schemes preserve the attractive properties of the usual DGM, such as the high order accuracy and compact stencil.