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Non-oscillatory entropy stable DG schemes for hyperbolic conservation law

Yuchang Liu, Wei Guo, Yan Jiang, Mengping Zhang

TL;DR

This work addresses the Gibbs phenomena and entropy-inequality violations that can arise in high-order DG discretizations of hyperbolic conservation laws. It introduces non-oscillatory entropy-stable DG schemes (NOES-DG) by embedding a local artificial viscosity term controlled by entropy variables, together with an integrating-factor SSP-RK time integrator to manage stiffness. The authors prove mass conservation, discrete entropy stability, and provide error estimates, while extending the formulation from 1D scalars to systems and 2D, and demonstrating robustness and high-order accuracy through extensive numerical tests on linear, Burgers, Buckley–Leverett, Euler, and jet problems. The approach also integrates a positivity-preserving limiter to ensure the physical admissibility of solutions in extreme cases, offering a simple, implementable DG framework with strong theoretical and practical performance. Overall, the NOES-DG method delivers entropy-consistent, oscillation-free solutions with high-order accuracy and practical positivity guarantees across a broad class of hyperbolic problems.

Abstract

In this paper, we propose a class of non-oscillatory, entropy-stable discontinuous Galerkin (NOES-DG) schemes for solving hyperbolic conservation laws. By incorporating a specific form of artificial viscosity, our new scheme directly controls entropy production and suppresses spurious oscillations. To address the stiffness introduced by the artificial terms, which can restrict severely time step sizes, we employ the integration factor strong stability-preserving Runge-Kutta method for time discretization. Furthermore, our method remains compatible with positivity-preserving limiters under suitable CFL conditions in extreme cases. Various numerical examples demonstrate the efficiency of the proposed scheme, showing that it maintains high-order accuracy in smooth regions and avoids spurious oscillations near discontinuities.

Non-oscillatory entropy stable DG schemes for hyperbolic conservation law

TL;DR

This work addresses the Gibbs phenomena and entropy-inequality violations that can arise in high-order DG discretizations of hyperbolic conservation laws. It introduces non-oscillatory entropy-stable DG schemes (NOES-DG) by embedding a local artificial viscosity term controlled by entropy variables, together with an integrating-factor SSP-RK time integrator to manage stiffness. The authors prove mass conservation, discrete entropy stability, and provide error estimates, while extending the formulation from 1D scalars to systems and 2D, and demonstrating robustness and high-order accuracy through extensive numerical tests on linear, Burgers, Buckley–Leverett, Euler, and jet problems. The approach also integrates a positivity-preserving limiter to ensure the physical admissibility of solutions in extreme cases, offering a simple, implementable DG framework with strong theoretical and practical performance. Overall, the NOES-DG method delivers entropy-consistent, oscillation-free solutions with high-order accuracy and practical positivity guarantees across a broad class of hyperbolic problems.

Abstract

In this paper, we propose a class of non-oscillatory, entropy-stable discontinuous Galerkin (NOES-DG) schemes for solving hyperbolic conservation laws. By incorporating a specific form of artificial viscosity, our new scheme directly controls entropy production and suppresses spurious oscillations. To address the stiffness introduced by the artificial terms, which can restrict severely time step sizes, we employ the integration factor strong stability-preserving Runge-Kutta method for time discretization. Furthermore, our method remains compatible with positivity-preserving limiters under suitable CFL conditions in extreme cases. Various numerical examples demonstrate the efficiency of the proposed scheme, showing that it maintains high-order accuracy in smooth regions and avoids spurious oscillations near discontinuities.

Paper Structure

This paper contains 16 sections, 4 theorems, 127 equations, 17 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Semi-discrete entropy stable DG schemes
  3. 1D scalar cases
  4. The classical DG discretization
  5. The entropy stable non-oscillatory DG schemes
  6. Theoretical properties
  7. 1D system
  8. Two-dimensional system
  9. Fully discretization
  10. Integral factor SSP-RK method
  11. Positivity-preserving limiter
  12. Numerical tests
  13. One-dimensional tests
  14. Two-dimensional tests
  15. Euler system
  16. ...and 1 more sections

Key Result

Theorem 2.1

(Mass Conservation) The NOES-DG scheme scheme1 is conservative, i.e.

Figures (17)

  • Figure 4.1: Example \ref{['ex:linear']}: one dimensional linear equation with non-smooth initial data. The numerical solution at $T = 2\pi$ with $N = 200$.
  • Figure 4.2: Example \ref{['ex:Burgers1D']}: One-dimensional Burgers' equation. The numerical solution at $T = 2.2$ with $N=200$ cells.
  • Figure 4.3: One-dimensional tests: Evolution of the total entropy with time.
  • Figure 4.4: Example \ref{['ex:BL1D']}: One dimensional Buckley-Leverett problem with different initial conditions. Numerical solutions at $T = 1$ with $N= 200$ cells.
  • Figure 4.5: Example \ref{['ex:BL1D']}: One dimensional Buckley-Leverett problem with different settings of $\sigma$. Numerical solutions at $T = 1$ with $N= 200$ cells.
  • ...and 12 more figures

Theorems & Definitions (28)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 18 more