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Convergence of a Hyperbolic Thermodynamically Compatible Finite Volume scheme for the Euler equations

Michael Dumbser, Mária Lukáčová-Medvid'ová, Andrea Thomann

TL;DR

The paper develops and analyzes a hyperbolic thermodynamically compatible finite-volume scheme for the compressible Euler equations, elevating entropy to a primary state variable and ensuring discrete energy conservation through the Abgrall flux. It provides a rigorous convergence framework in the dissipative weak-solutions setting, requiring uniform lower bounds on density and upper bounds on energy to obtain stability, consistency, and weak convergence (via Young measures). When a strong solution exists, the scheme converges strongly; otherwise, Cesàro averages and the first variance converge to a dissipative weak solution, with numerical KH tests illustrating the convergence behavior. The work bridges thermodynamic structure, discrete energy conservation, and rigorous convergence analysis for nonlinear hyperbolic systems, with potential impact on robust, energy-consistent simulations of compressible flows.

Abstract

We study the convergence of a novel family of thermodynamically compatible schemes for hyperbolic systems (HTC schemes) in the framework of dissipative weak solutions, applied to the Euler equations of compressible gas dynamics. Two key novelties of our method are i) entropy is treated as one of the main field quantities and ii) the total energy conservation is a consequence of compatible discretization and application of the Abgrall flux.

Convergence of a Hyperbolic Thermodynamically Compatible Finite Volume scheme for the Euler equations

TL;DR

The paper develops and analyzes a hyperbolic thermodynamically compatible finite-volume scheme for the compressible Euler equations, elevating entropy to a primary state variable and ensuring discrete energy conservation through the Abgrall flux. It provides a rigorous convergence framework in the dissipative weak-solutions setting, requiring uniform lower bounds on density and upper bounds on energy to obtain stability, consistency, and weak convergence (via Young measures). When a strong solution exists, the scheme converges strongly; otherwise, Cesàro averages and the first variance converge to a dissipative weak solution, with numerical KH tests illustrating the convergence behavior. The work bridges thermodynamic structure, discrete energy conservation, and rigorous convergence analysis for nonlinear hyperbolic systems, with potential impact on robust, energy-consistent simulations of compressible flows.

Abstract

We study the convergence of a novel family of thermodynamically compatible schemes for hyperbolic systems (HTC schemes) in the framework of dissipative weak solutions, applied to the Euler equations of compressible gas dynamics. Two key novelties of our method are i) entropy is treated as one of the main field quantities and ii) the total energy conservation is a consequence of compatible discretization and application of the Abgrall flux.

Paper Structure

This paper contains 19 sections, 9 theorems, 114 equations, 3 figures, 1 table.

Table of Contents

  1. Abstract
  2. Keywords
  3. Introduction
  4. The compressible Euler equations
  5. Inviscid equations
  6. Vanishing viscosity regularization
  7. The numerical scheme
  8. Spatial discretization
  9. The hyperbolic thermodynamically compatible FV scheme
  10. Upper and lower bounds of conserved and thermodynamical variables
  11. Stability of the Numerical Scheme
  12. Consistency
  13. Consistency errors
  14. Consistency Formulation
  15. Convergence analysis via dissipative weak solutions
  16. ...and 4 more sections

Key Result

Lemma 3.2

Let $T_K (t) > 0$ uniformly and $({\mathbf{H}}_h)_\sigma$ given by eq:Roe_matrix. Consider the modified numerical flux eq:HTC_num_flux_q given by where the constant $\overline{\kappa}$ being an upper bound on the eigenvalues of $~{\mathbf{H}}$. Then the coefficient $\alpha_\sigma \left({{\mathbf{H}}}_h\right)_\sigma + \left( |\alpha_\sigma| \overline{\kappa} + \lambda_\sigma\right) \mathbb{I}$ in

Figures (3)

  • Figure 1: Kelvin-Helmholtz problem: Density computed with the HTC scheme at time $T=2$ for different meshes with $n \times n$ cells.
  • Figure 2: Kelvin-Helmholtz problem: Cesáro averages $\tilde{U}_{h_n}$ of the density computed with the HTC scheme at time $T=2$ on meshes with $n \times n$ cells, starting from $n=256$.
  • Figure 3: Kelvin-Helmholtz problem: First variance $U_{h_n}^\dagger$ of the density computed with the HTC scheme at time $T=2$ starting from the $n=256$ mesh.

Theorems & Definitions (19)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.5: Estimate of the entropy production term
  • proof
  • Lemma 3.6
  • proof
  • Definition 4.1: weak BV condition
  • ...and 9 more