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A limiter-based approach to construct high-order fully-discrete entropy stable explicit DG schemes for hyperbolic conservation laws

Yuchang Liu, Wei Guo, Yan Jiang, Zheng Sun

Abstract

This paper presents a class of novel high-order fully-discrete entropy stable (ES) discontinuous Galerkin (DG) schemes with explicit time discretization. The proposed methodology exploits a critical observation from [4] that the cell averages of classical DG solutions with forward Euler time stepping satisfy an ``entropy-stable-like'' property. Building on this result, fully-discrete entropy stability is rigorously enforced through a simple Zhang--Shu-type scaling limiter [45] applied as a post-processing step, without modifying the underlying spatial discretization. Furthermore, the proposed methodology can simultaneously enforce multiple cell entropy inequalities, a capability unavailable in existing ES DG schemes. High-order accuracy in time is achieved by using strong-stability-preserving (SSP) multistep methods. Theoretically, we prove that the proposed scheme indeed maintains high-order accuracy and establish a Lax--Wendroff-type theorem guaranteeing that the limit of the numerical solutions, if it exists, satisfies the desired entropy inequality. Extensive numerical tests for scalar equations and systems, including the nonconvex Buckley--Leverett problem and extreme examples of Euler equations, demonstrate optimal accuracy, enforcement of multiple entropy conditions, and strong robustness.

A limiter-based approach to construct high-order fully-discrete entropy stable explicit DG schemes for hyperbolic conservation laws

Abstract

This paper presents a class of novel high-order fully-discrete entropy stable (ES) discontinuous Galerkin (DG) schemes with explicit time discretization. The proposed methodology exploits a critical observation from [4] that the cell averages of classical DG solutions with forward Euler time stepping satisfy an ``entropy-stable-like'' property. Building on this result, fully-discrete entropy stability is rigorously enforced through a simple Zhang--Shu-type scaling limiter [45] applied as a post-processing step, without modifying the underlying spatial discretization. Furthermore, the proposed methodology can simultaneously enforce multiple cell entropy inequalities, a capability unavailable in existing ES DG schemes. High-order accuracy in time is achieved by using strong-stability-preserving (SSP) multistep methods. Theoretically, we prove that the proposed scheme indeed maintains high-order accuracy and establish a Lax--Wendroff-type theorem guaranteeing that the limit of the numerical solutions, if it exists, satisfies the desired entropy inequality. Extensive numerical tests for scalar equations and systems, including the nonconvex Buckley--Leverett problem and extreme examples of Euler equations, demonstrate optimal accuracy, enforcement of multiple entropy conditions, and strong robustness.

Paper Structure

This paper contains 21 sections, 22 theorems, 173 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Entropy analysis
  3. Fully-discrete explicit entropy stable scheme for scalar problems
  4. DG formulation
  5. Forward Euler ES scheme
  6. High-order ES scheme
  7. Multiple entropy inequality
  8. Fully-discrete explicit entropy stable scheme for systems
  9. DG formulation
  10. Forward Euler ES scheme
  11. High-order ES scheme
  12. Compatibility with other limiters
  13. Numerical tests
  14. Scalar problems
  15. One-dimensional Euler systems
  16. ...and 6 more sections

Key Result

Lemma 2.2

\newlabellem:mock0 A strictly convex function $\mathcal{U}$ serves as an entropy function of eq:hcl-d if and only if eq:hcl-v is a symmetrization of eq:hcl-d.

Figures (12)

  • Figure 1: Example \ref{['ex:linear']}: Linear equation. The $(x,t)$-distribution of limited cells of ES1 scheme with $N_x=80$.
  • Figure 2: Example \ref{['ex:burgers']}: One-dimensional Burgers' equation. The cell entropy inequality violation and evolution of total entropy with time. $\mathbb P^2$ approximation are used with $N_x=80$.
  • Figure 3: Example \ref{['ex:BL']}: Buckley--Leverett equation. Numerical solution of different entropy functions at $T = 1$ with $N_x=80$.
  • Figure 4: Example \ref{['ex:Sod']}: Sod shock tube. The result at $T=0.4$ with $N_x=200$.
  • Figure 5: Example \ref{['ex:Shuosher']}: Shu--Osher problem. The result at $T=1.8$ with $N_x=200$.
  • ...and 7 more figures

Theorems & Definitions (58)

  • Definition 2.1: Entropy function
  • Lemma 2.2: Mock
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Proof 1
  • Lemma 3.6
  • Proposition 3.7
  • ...and 48 more