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A new class of entropy stable fluctuations for the discontinuous Galerkin method with application to the Saint-Venant-Exner model

Patrick Ersing, Andrew R. Winters

TL;DR

This work addresses stability challenges in high-order DG methods for nonconservative hyperbolic systems by introducing two general entropy-conservative fluctuations (EC) and a novel blending approach to obtain entropy-stable (ES) dissipation. The authors develop a flux-differencing DGSEM that preserves the entropy inequality semi-discretely, even in the presence of nonconservative products, and apply it to the Saint-Venant-Exner system with a well-balanced formulation. They present two EC-fluctuation strategies—one via a linear path in entropy variables and another via a closed-form two-point flux—along with a blending operator that combines Roe- and LLF-like dissipation to ensure stability without sacrificing steady states. Numerical experiments demonstrate fourth-order convergence, entropy conservation for EC fluctuations, robust entropy stability with blended dissipation, and exact lake-at-rest preservation for various polynomial orders, highlighting the method’s practical potential for complex geophysical flows. The work provides a general, model-independent framework for ES nonconservative DG methods and a reproducible SVE implementation for broader adoption.

Abstract

In this work we consider entropy stable discontinuous Galerkin methods applied to nonconservative hyperbolic systems. We introduce a new class of entropy conservative fluctuations that allow us to construct entropy conservative schemes without any system-specific derivations. We demonstrate that a loss of entropy symmetrization for nonconservative systems restricts the design of entropy stable fluctuations and propose a novel blending procedure to construct entropy stable dissipation terms from general numerical viscosity matrices. The resulting methodology is applied to develop a high-order, entropy stable, and well-balanced approximation for the Saint-Venant-Exner system. Numerical tests are presented to verify the theoretical findings and demonstrate the performance and robustness of the proposed scheme.

A new class of entropy stable fluctuations for the discontinuous Galerkin method with application to the Saint-Venant-Exner model

TL;DR

This work addresses stability challenges in high-order DG methods for nonconservative hyperbolic systems by introducing two general entropy-conservative fluctuations (EC) and a novel blending approach to obtain entropy-stable (ES) dissipation. The authors develop a flux-differencing DGSEM that preserves the entropy inequality semi-discretely, even in the presence of nonconservative products, and apply it to the Saint-Venant-Exner system with a well-balanced formulation. They present two EC-fluctuation strategies—one via a linear path in entropy variables and another via a closed-form two-point flux—along with a blending operator that combines Roe- and LLF-like dissipation to ensure stability without sacrificing steady states. Numerical experiments demonstrate fourth-order convergence, entropy conservation for EC fluctuations, robust entropy stability with blended dissipation, and exact lake-at-rest preservation for various polynomial orders, highlighting the method’s practical potential for complex geophysical flows. The work provides a general, model-independent framework for ES nonconservative DG methods and a reproducible SVE implementation for broader adoption.

Abstract

In this work we consider entropy stable discontinuous Galerkin methods applied to nonconservative hyperbolic systems. We introduce a new class of entropy conservative fluctuations that allow us to construct entropy conservative schemes without any system-specific derivations. We demonstrate that a loss of entropy symmetrization for nonconservative systems restricts the design of entropy stable fluctuations and propose a novel blending procedure to construct entropy stable dissipation terms from general numerical viscosity matrices. The resulting methodology is applied to develop a high-order, entropy stable, and well-balanced approximation for the Saint-Venant-Exner system. Numerical tests are presented to verify the theoretical findings and demonstrate the performance and robustness of the proposed scheme.
Paper Structure (19 sections, 8 theorems, 92 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 19 sections, 8 theorems, 92 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Entropy analysis
  3. Entropy stable DGSEM
  4. Nonconservative DGSEM
  5. Entropy conservative DGSEM
  6. Entropy conservative fluctuations: Strategy 1
  7. Entropy conservative fluctuations: Strategy 2
  8. Entropy stable DGSEM
  9. Saint-Venant-Exner system
  10. Numerical method
  11. Well-balancedness
  12. Results
  13. Convergence test
  14. Channel flow problem
  15. Entropy conservation
  16. ...and 4 more sections

Key Result

Lemma 1

The flux differencing volume integral eq:flux_diff_volume_integral with fluctuations satisfying eq:fluctuation_condition_c approximates the nonconservative product $\underline{A}\mathbf u_{\xi}$ with the same order of accuracy $p$ as the derivative matrix eq:definition_derivative_matrix.

Figures (4)

  • Figure 1: Time evolution of the sediment height $b$ in the channel flow test case, showing the approximate reference solution and numerical results obtained with $\mathbb{P}^4$ polynomials using EC fluctuations $\mathbf D_{EC,1}$ and $\mathbf D_{EC,2}$.
  • Figure 2: Time evolution of the sediment height $b$ in the channel flow test case, showing the approximate reference solution and numerical solutions results with $\mathbb{P}^4$ polynomials and ES fluctuations at interfaces.
  • Figure 3: Total entropy over time for the channel flow test case with different ES fluctuations at interfaces.
  • Figure 4: Water and sediment height obtained with ES fluctuations $\mathbf D_{EC,2} + \underline{Q}_{llf}$ (left) and $\mathbf D_{EC,2} + \underline{Q}_{ES-roe}$ (right) for the well-balanced test case at final time $t=10$. Results are shown for polynomial degrees $\mathbb{P}^0$, $\mathbb{P}^1$, and $\mathbb{P}^2$.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • remark 1
  • definition 1: EC fluctuation
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 9 more