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A posteriori analysis for nonlinear convection-diffusion systems

Andreas Dedner, Jan Giesselmann, Kiwoong Kwon, Tristan Pryer

TL;DR

This work provides reliable a posteriori error estimators for RKdG discretizations of nonlinear convection-diffusion systems, focusing on convection-dominated and degenerate parabolic regimes. The authors develop a space-time reconstruction approach and a general relative entropy framework to derive computable upper bounds that hold for both explicit and IMEX time stepping across broad flux and diffusion models. The estimators are shown to be efficient, with the error converging at the same rate as the estimators, and the analysis explicitly tracks the dependence on viscosity and diffusion degeneracy. Numerical experiments across linear and nonlinear scalar and system cases validate robustness, optimality, and the viscosity-independence of the indicators, demonstrating practical applicability for challenging convection-dominated flows.

Abstract

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

A posteriori analysis for nonlinear convection-diffusion systems

TL;DR

This work provides reliable a posteriori error estimators for RKdG discretizations of nonlinear convection-diffusion systems, focusing on convection-dominated and degenerate parabolic regimes. The authors develop a space-time reconstruction approach and a general relative entropy framework to derive computable upper bounds that hold for both explicit and IMEX time stepping across broad flux and diffusion models. The estimators are shown to be efficient, with the error converging at the same rate as the estimators, and the analysis explicitly tracks the dependence on viscosity and diffusion degeneracy. Numerical experiments across linear and nonlinear scalar and system cases validate robustness, optimality, and the viscosity-independence of the indicators, demonstrating practical applicability for challenging convection-dominated flows.

Abstract

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

Paper Structure

This paper contains 24 sections, 19 theorems, 195 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Numerical schemes and reconstructions
  3. RKdG schemes
  4. Reconstruction techniques
  5. Decomposing the residual
  6. General Relative Entropy Estimates
  7. Linear scalar problems
  8. Non-linear scalar problems
  9. Linear Systems
  10. The linear wave equation with degenerate diffusion
  11. The non-linear wave equation with degenerate diffusion
  12. Numerical experiments
  13. Linear advection-diffusion equation
  14. Convergence rates of the RKdG solution
  15. Optimality and robustness of the a posteriori estimators
  16. ...and 9 more sections

Key Result

Lemma 2.3

The reconstruction $\widehat{\boldsymbol{u}}_h^t$ as given in Definition def:grec is well-defined, computable, and belongs to the space $\operatorname{W}^{1,\infty}(0,T; \mathbb V_q^s)$.

Figures (13)

  • Figure 1: Two-stage reconstruction methodology. (Left) Discrete solution $\boldsymbol{u}^n_h$ at time nodes. (Center) Temporal reconstruction $\widehat{\boldsymbol{u}}^t_h$. (Right) Space-time reconstruction $\widehat{\boldsymbol{u}}^{ts}$.
  • Figure 2: RKdG solution error in $\operatorname L\xspace_{\infty}(0,T;\operatorname L\xspace_{2}(\Omega\xspace))$-norm for linear advection-diffusion equation \ref{['eq:linear-adv-diff']} for polynomial degrees $q=1$ (left) and $q=2$ (right)
  • Figure 3: RKdG error in dG energy norm for linear advection-diffusion equation \ref{['eq:linear-adv-diff']} for polynomial degrees $q=1$ (left) and $q=2$ (right)
  • Figure 4: Residual norm $\left\|r_1\right\|_{\operatorname L\xspace_{1}(0,T;\operatorname L\xspace_{2}(\Omega\xspace))}$ for linear advection-diffusion equation \ref{['eq:linear-adv-diff']} for polynomial degrees $q=1$ (left) and $q=2$ (right)
  • Figure 5: Residual norm $\left\|r_2\right\|_{\operatorname L\xspace_{2}(0,T;\operatorname H\xspace^{-1}(\Omega\xspace))}$ for linear advection-diffusion equation \ref{['eq:linear-adv-diff']} for polynomial degrees $q=1$ (left) and $q=2$ (right)
  • ...and 8 more figures

Theorems & Definitions (63)

  • Definition 2.1: dG energy norm
  • Definition 2.2: Temporal reconstruction for third order schemes
  • Lemma 2.3: Properties of reconstruction in time DG16
  • Remark 2.5: Admissible numerical fluxes
  • Definition 2.6: Space-time reconstruction
  • Lemma 2.7: Properties of space-time reconstruction DG16
  • Remark 2.8: Regularity of space-time reconstruction
  • Remark 2.9: Regularity of the residual
  • Remark 2.10: Splitting the residual
  • Remark 3.1: Entropy
  • ...and 53 more