Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods

Abstract

One of the beneficial properties of the discontinuous Galerkin method is the accurate wave propagation properties. That is, the semi-discrete error has dissipation errors of order $2k+1$ ($\le Ch^{2k+1}$) and order $2k+2$ for dispersion ($\le Ch^{2k+2}$). Previous studies have concentrated on the order of accuracy, and neglected the important role that the error constant, C,  plays in these estimates. In this article, we show the important role of the error constant in the dispersion and dissipation error for discontinuous Galerkin approximation of polynomial degree k,  where $k=0,1,2,3.$ This gives insight into why one may want a more centred flux for a piecewise constant or quadratic approximation than for a piecewise linear or cubic approximation. We provide an explicit formula for these error constants. This is illustrated through one particular flux, the upwind-biased flux introduced by Meng et al., as it is a convex combination of the upwind and downwind fluxes. The studies of wave propagation are typically done through a Fourier ansatz. This higher order Fourier information can be extracted using the smoothness-increasing accuracy-conserving (SIAC) filter. The SIAC filter ties the higher order Fourier information to the negative-order norm in physical space. We show that both the proofs of the ability of the SIAC filter to extract extra accuracy and numerical results are unaffected by the choice of flux.

Funding

Air Force Office of Scientific Research(FA8655-09-1-3017)