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On Local Minimum Entropy Principle of High-Order Schemes for Relativistic Euler Equations

Shumo Cui, Kailiang Wu, Linfeng Xu

TL;DR

This work establishes the minimum entropy principle (MEP) for the special relativistic Euler equations with a broad class of Synge-type equations of state, proving a convexity condition on a generated entropy pair and deriving both local and global MEP results. It then develops a rigorous, entropy-preserving high-order numerical framework based on geometric quasi-linearization (GQL) to linearize nonlinear entropy constraints, enabling provably locally entropy-preserving DG/finite-volume schemes in 1D and multidimensional settings. Central innovations include two a priori estimators for local entropy lower bounds and a limiting procedure to enforce entropy bounds at high order without sacrificing accuracy, demonstrated through extensive 1D/2D tests across several EOSs. The paper shows that locally enforcing the MEP yields superior oscillation control and preserves high-order accuracy, offering a robust, extensible approach for relativistic hydrodynamics and potentially other models admitting the MEP. These results provide a principled path toward entropy-stable, high-fidelity simulations in relativistic flows and related systems.

Abstract

This paper establishes the minimum entropy principle (MEP) for the relativistic Euler equations with a broad class of equations of state (EOSs) and addresses the challenge of preserving the local version of the discovered MEP in high-order numerical schemes. At the continuous level, we find out a family of entropy pairs for the relativistic Euler equations and provide rigorous analysis to prove the strict convexity of entropy under a necessary and sufficient condition. At the numerical level, we develop a rigorous framework for designing provably entropy-preserving high-order schemes that ensure both physical admissibility and the discovered MEP. The relativistic effects, coupled with the abstract and general EOS formulation, introduce significant challenges not encountered in the nonrelativistic case or with the ideal EOS. In particular, entropy is a highly nonlinear and implicit function of the conservative variables, making it particularly difficult to enforce entropy preservation. To address these challenges, we establish a series of auxiliary theories via highly technical inequalities. Another key innovation is the use of geometric quasi-linearization (GQL), which reformulates the nonlinear constraints into equivalent linear ones by introducing additional free parameters. These advancements form the foundation of our entropy-preserving analysis. We propose novel, robust, locally entropy-preserving high-order frameworks. A central challenge is accurately estimating the local minimum of entropy, particularly in the presence of shock waves at unknown locations. To address this, we introduce two new approaches for estimating local lower bounds of specific entropy, which prove effective for both smooth and discontinuous problems. Numerical experiments demonstrate that our entropy-preserving methods maintain high-order accuracy while effectively suppressing spurious oscillations.

On Local Minimum Entropy Principle of High-Order Schemes for Relativistic Euler Equations

TL;DR

This work establishes the minimum entropy principle (MEP) for the special relativistic Euler equations with a broad class of Synge-type equations of state, proving a convexity condition on a generated entropy pair and deriving both local and global MEP results. It then develops a rigorous, entropy-preserving high-order numerical framework based on geometric quasi-linearization (GQL) to linearize nonlinear entropy constraints, enabling provably locally entropy-preserving DG/finite-volume schemes in 1D and multidimensional settings. Central innovations include two a priori estimators for local entropy lower bounds and a limiting procedure to enforce entropy bounds at high order without sacrificing accuracy, demonstrated through extensive 1D/2D tests across several EOSs. The paper shows that locally enforcing the MEP yields superior oscillation control and preserves high-order accuracy, offering a robust, extensible approach for relativistic hydrodynamics and potentially other models admitting the MEP. These results provide a principled path toward entropy-stable, high-fidelity simulations in relativistic flows and related systems.

Abstract

This paper establishes the minimum entropy principle (MEP) for the relativistic Euler equations with a broad class of equations of state (EOSs) and addresses the challenge of preserving the local version of the discovered MEP in high-order numerical schemes. At the continuous level, we find out a family of entropy pairs for the relativistic Euler equations and provide rigorous analysis to prove the strict convexity of entropy under a necessary and sufficient condition. At the numerical level, we develop a rigorous framework for designing provably entropy-preserving high-order schemes that ensure both physical admissibility and the discovered MEP. The relativistic effects, coupled with the abstract and general EOS formulation, introduce significant challenges not encountered in the nonrelativistic case or with the ideal EOS. In particular, entropy is a highly nonlinear and implicit function of the conservative variables, making it particularly difficult to enforce entropy preservation. To address these challenges, we establish a series of auxiliary theories via highly technical inequalities. Another key innovation is the use of geometric quasi-linearization (GQL), which reformulates the nonlinear constraints into equivalent linear ones by introducing additional free parameters. These advancements form the foundation of our entropy-preserving analysis. We propose novel, robust, locally entropy-preserving high-order frameworks. A central challenge is accurately estimating the local minimum of entropy, particularly in the presence of shock waves at unknown locations. To address this, we introduce two new approaches for estimating local lower bounds of specific entropy, which prove effective for both smooth and discontinuous problems. Numerical experiments demonstrate that our entropy-preserving methods maintain high-order accuracy while effectively suppressing spurious oscillations.

Paper Structure

This paper contains 22 sections, 21 theorems, 212 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Minimum Entropy Principle
  3. General Equation of State
  4. Convex Entropy Pairs
  5. Sufficiency
  6. Necessity
  7. Minimum Principle on Specific Entropy
  8. Crucial Inequalities for Entropy-Preserving Analysis
  9. Analysis and Design of Entropy-Preserving Schemes
  10. Geometric Quasi-Linearization (GQL): Overcoming the Challenges Arising from Nonlinearity
  11. 1D Entropy-Preserving Schemes
  12. First-Order 1D Scheme
  13. High-Order 1D Schemes
  14. Limiters
  15. Multidimensional Entropy-Preserving Schemes
  16. ...and 7 more sections

Key Result

Lemma 2.1

\newlabellem:convexentropy0 For a general EOS satisfying condition eq:122 and $h(\theta) - \theta h'(\theta) \neq 0$, the function $\mathcal{E}(\mathbf{U})$ defined in eq:409 is strictly convex if and only if the following conditions hold: where $\mathcal{G}$ denotes the physically admissible state set defined as

Figures (8)

  • Figure 1: Example \ref{['Ex5.1.1']}: $l^1$, $l^2$, and $l^\infty$ errors in $\rho$ for locally entropy-preserving DG methods on various mesh resolutions, comparing different entropy bound estimation approaches \ref{['LV Final star']}, \ref{['CJK']}, \ref{['eq:3089']}, and \ref{['1DMPrelax']}.
  • Figure 2: Example \ref{['Ex5.1.2']}: the rest-mass density $\rho$ obtained by the $\mathbb{P}^3$-based DG methods with and without different entropy limiters (red circles) and reference solution (black solid line), $t = 0.4$, $\Delta x = 1/400$.
  • Figure 3: Example \ref{['Ex5.1.3']}: the rest-mass density $\rho$ obtained by the $\mathbb{P}^3$-based DG methods without or with entropy-preserving limiters (red circles) and reference solution (black solid line), $t = 0.4$, $\Delta x = 1/400$.
  • Figure 4: Example \ref{['Ex5.2.1']}: $l^1$, $l^2$, and $l^\infty$ errors in $\rho$ for locally entropy-preserving DG methods on various mesh resolutions, comparing different entropy bound estimation approaches \ref{['LV Final star']}, \ref{['CJK']}, \ref{['2D_CUI']}, and \ref{['2DMPrelax']}.
  • Figure 5: Example \ref{['Ex5.2.2']}: Contour plots of $\log_{10}\rho$ at $t=0.8$ obtained using $\mathbb{P}^3$-based DG methods without or with entropy-preserving limiters on a uniform mesh with $\Delta x = \Delta y = 1/125$. 18 equally spaced contour lines are displayed.
  • ...and 3 more figures

Theorems & Definitions (57)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Theorem 2.3
  • Proof 3
  • Theorem 2.4
  • Lemma 3.1
  • Proof 4
  • Lemma 3.2
  • ...and 47 more