Relaxing the CFL Number of the Discontinuous Galerkin Method

Abstract

We propose a family of high order methods for the 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 semidiscrete scheme are related to the $[\frac{p}{p+1}]$ Pade approximants of $\exp(z)$ and $\exp(-z)$. These Pade approximants are responsible for the superconvergent $\mathcal{O}(h^{2p+2})$ and $\mathcal{O}(h^{2p+1})$ errors in dispersion and dissipation, respectively, and the restriction of the CFL number when increasing the order of approximation, $p$. By modifying the DG method 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 DG method, while having the same order of convergence in the $\mathcal{L}^2$ norm. This is achieved through modifications to the numerical flux. The schemes preserve the attract...