2016/02/20 by Daniel Frean, Frean, Daniel, Jennifer K. Ryan +2
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1602.06444
openalex publication_date 2016/02/20 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
In this paper we investigate the superconvergence properties of the\ndiscontinuous Galerkin method based on the upwind-biased flux for linear\ntime-dependent hyperbolic equations. We prove that for even-degree polynomials,\nthe method is locally \O(hk+2) superconvergent at roots of a\nlinear combination of the left- and right-Radau polynomials. This linear\ncombination depends on the value of \θ used in the flux. For odd-degree\npolynomials, the scheme is superconvergent provided that a proper global\ninitial interpolation can be defined. We demonstrate numerically that, for\ndecreasing \θ, the discretization errors decrease for even polynomials\nand grow for odd polynomials. We prove that the use of Smoothness-Increasing\nAccuracy-Conserving (SIAC) filters is still able to draw out the\nsuperconvergence information and create a globally smooth and superconvergent\nsolution of \O(h2k+1) for linear hyperbolic equations. Lastly, we\nbriefly consider the spectrum of the upwind-biased DG operator and demonstrate\nthat the price paid for the introduction of the parameter \θ is limited\nto a contribution to the constant attached to the post-processed error term.\n