A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems

TL;DR

This paper introduces reliable a posteriori error estimators for Runge-Kutta discontinuous Galerkin methods with SIAC post-processing applied to nonlinear convection-diffusion systems, especially in convection-dominated regimes.

Contribution

It develops error estimators that are uniform in the vanishing viscosity limit and leverages SIAC filtering for superconvergence in complex multidimensional settings.

Findings

Error bounds converge with the same order as the reconstructed solution's error.

Estimators are reliable and uniform in the vanishing viscosity limit.

Numerical experiments confirm the effectiveness of the error bounds.

Abstract

We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.