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

Jan Giesselmann, Kiwoong Kwon, Sebastian Krumscheid

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.

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

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.

Paper Structure

This paper contains 19 sections, 4 theorems, 87 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discretization
  4. Discontinuous Galerkin spatial discretization
  5. Runge--Kutta time discretization
  6. Space-time reconstruction
  7. Temporal reconstruction
  8. Spatial reconstruction by SIAC filtering
  9. Residual splitting for the space-time reconstruction
  10. Stability estimates and a posteriori analysis
  11. Linear advection--diffusion equation
  12. Nonlinear scalar equation
  13. Diffusive p-system
  14. Numerical experiments
  15. Linear advection--diffusion equation
  16. ...and 4 more sections

Key Result

Lemma 4.7

The space-time reconstruction $\widehat{\mathbf{u}}^{ts}$ satisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Piecewise polynomial structure of the SIAC-filtered function for $v_h(y)=\chi_{[0,h]}(y)$ on an equidistant mesh. Panel (a) shows $v_h$. Panels (b) and (c) show the SIAC kernel $K_h^{(2q+1,q+1)}$ and the SIAC-filtered function $K_h^{(2q+1,q+1)}*v_h$ for $q=1$ and $q=2$, respectively. The open circles mark the points where the kernel is not smooth, and the filled circles mark the points where the filtered function is not smooth. For $q=1$, the $x$-coordinates of the filled circles coincide with the original mesh nodes, whereas for $q=2$ they coincide with the midpoints of cells of the original mesh.

Theorems & Definitions (16)

  • Definition 2.1: Weak solution and entropy weak solution
  • Remark 2.2: Well-posedness
  • Definition 4.1: Central B-spline 2014Ryan
  • Definition 4.2: SIAC kernel 2014Ryan
  • Definition 4.3: SIAC filtering
  • Remark 4.4: Kernel support
  • Remark 4.5: Piecewise polynomial structure of SIAC filtering
  • Remark 4.6: Superconvergence properties of SIAC filtering
  • Lemma 4.7: Regularity of space-time reconstruction
  • proof
  • ...and 6 more