Superconvergence properties of an upwind-biased discontinuous Galerkin method

TL;DR

This paper studies the superconvergence properties of an upwind-biased discontinuous Galerkin method for hyperbolic equations, proving superconvergence at specific points and with filters, and analyzing the spectral effects of the flux parameter.

Contribution

It provides theoretical proofs of superconvergence for even and odd polynomial degrees and demonstrates the effectiveness of SIAC filters in enhancing solution accuracy.

Findings

Superconvergence at roots of combined Radau polynomials for even degrees.

Superconvergence depends on the flux parameter  in odd degrees.

SIAC filters can achieve  h^{2k+1} superconvergence.

Abstract

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method based on the upwind-biased flux for linear time-dependent hyperbolic equations. We prove that for even-degree polynomials, the method is locally $\mathcal{O}(h^{k+2})$ superconvergent at roots of a linear combination of the left- and right-Radau polynomials. This linear combination depends on the value of $\theta$ used in the flux. For odd-degree polynomials, the scheme is superconvergent provided that a proper global initial interpolation can be defined. We demonstrate numerically that, for decreasing $\theta$, the discretization errors decrease for even polynomials and grow for odd polynomials. We prove that the use of Smoothness-Increasing Accuracy-Conserving (SIAC) filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution of $\mathcal{O}(h^{2k+1})$ for linear hyperbolic equations. Lastly, we briefly consider the spectrum of the upwind-biased DG operator and demonstrate that the price paid for the introduction of the parameter $\theta$ is limited to a contribution to the constant attached to the post-processed error term.