A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems
This work provides incremental improvements in error analysis for computational fluid dynamics simulations, specifically benefiting researchers and engineers dealing with convection-dominated systems.
The paper tackled the problem of developing reliable a posteriori error estimators for Runge-Kutta discontinuous Galerkin schemes applied to nonlinear convection-diffusion systems, achieving error bounds with constants uniform in the vanishing viscosity limit and showing in numerical experiments that these bounds converge at the same order as the error of the reconstructed solution.
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.