Superconvergence properties of an upwind-biased discontinuous Galerkin method
This work provides theoretical and numerical insights into the superconvergence behavior of a parameterized DG method, which is incremental for researchers in numerical analysis of hyperbolic PDEs.
The paper proves superconvergence properties of the upwind-biased discontinuous Galerkin method for linear hyperbolic equations, showing local O(h^{k+2}) superconvergence at Radau points for even-degree polynomials and global O(h^{2k+1}) superconvergence via SIAC filters. Numerical results demonstrate that decreasing the flux parameter θ reduces errors for even polynomials but increases them for odd polynomials.
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 $θ$ 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 $θ$, 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 $θ$ is limited to a contribution to the constant attached to the post-processed error term.