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Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

Shaoshuai Chu, Michael Herty, Alexander Kurganov, Maria Lukacova-Medvidova, Changsheng Yu

TL;DR

This work investigates dissipative weak (DW) solutions to the 2-D Euler equations for gas dynamics by applying a spectrum of high-order numerical schemes—LCDCU, LDCU, and VFV—with A-WENO enhancements. By examining 2-D Riemann problems and Kelvin-Helmholtz instability, the authors show that different schemes may converge to distinct DW solutions, while Cesàro averages over orders exhibit strong convergence to a DW limit and reveal scheme-dependent Young measures. They quantify convergence via entropy production and energy defects to inform selection criteria among DW solutions. The findings highlight the nonuniqueness of DW limits in turbulent-like regimes and propose principled criteria for selecting physically relevant solutions, with implications for turbulence modeling and measure-valued PDE analysis.

Abstract

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of ${\cal K}$-convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.

Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

TL;DR

This work investigates dissipative weak (DW) solutions to the 2-D Euler equations for gas dynamics by applying a spectrum of high-order numerical schemes—LCDCU, LDCU, and VFV—with A-WENO enhancements. By examining 2-D Riemann problems and Kelvin-Helmholtz instability, the authors show that different schemes may converge to distinct DW solutions, while Cesàro averages over orders exhibit strong convergence to a DW limit and reveal scheme-dependent Young measures. They quantify convergence via entropy production and energy defects to inform selection criteria among DW solutions. The findings highlight the nonuniqueness of DW limits in turbulent-like regimes and propose principled criteria for selecting physically relevant solutions, with implications for turbulence modeling and measure-valued PDE analysis.

Abstract

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of -convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.
Paper Structure (21 sections, 3 theorems, 56 equations, 15 figures, 4 tables)

This paper contains 21 sections, 3 theorems, 56 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Consistent Approximations and Dissipative Weak Solutions
  3. Numerical Schemes
  4. Low-Order FV Schemes
  5. High-Order FD A-WENO Schemes
  6. $r=3$ (third-order schemes)
  7. $r=5$ (fifth-order schemes)
  8. $r=7$ (seventh-order schemes)
  9. $r=9$ (ninth-order schemes)
  10. Numerical Results
  11. 2-D Riemann Problems
  12. Configuration 2.
  13. Configuration 4.
  14. Configuration 3.
  15. Kelvin-Helmholtz (KH) Instability
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let $\{\rho_\ell,\bm m_\ell,S_\ell\}_{\ell=1}^\infty$ be a family of consistent approximations of (1.1)--(1.4) in the sense of Definition D3, such that Suppose the limit $(\rho,\bm m,S)$ is a weak solution of (1.1)--(1.4). Then, there exists a subsequence $\{\rho_{\ell_k},\bm m_{\ell_k},S_{\ell_k}\}$ such that

Figures (15)

  • Figure 4.1: Configuration 2: Density computed by the first- (top row), third- (middle row), and ninth-order (bottom row) LCDCU (left column), LDCU (middle column), and VFV (right column) schemes.
  • Figure 4.2: Configuration 4: Density computed by the first- (top row), third- (middle row), and ninth-order (bottom row) LCDCU (left column), LDCU (middle column), and VFV (right column) schemes.
  • Figure 4.3: Configuration 3: Density computed by the first- (top row), third- (middle row), and seventh-order (bottom row) LCDCU (left column), LDCU (middle column), and VFV (right column) schemes.
  • Figure 4.4: Configuration 3: $\widetilde{\rho}_3$ (top row), $\widetilde{\rho}_4$ (middle row), and $\widetilde{\rho}_5$ (bottom row) computed by the LCDCU (left column), LDCU (middle column), and VFV (right column) schemes.
  • Figure 4.5: Configuration 3: $\rho^T_1$ (top row), $\rho^T_3$ (middle row), and $\rho^T_5$ (bottom row) computed by the LCDCU (left column), LDCU (middle column), and VFV (right column) schemes.
  • ...and 10 more figures

Theorems & Definitions (7)

  • Definition 2.1: Consistent approximation
  • Theorem 2.2
  • Definition 2.3: DW solution
  • Remark 2.1
  • Theorem 2.4: Weak versus strong convergence
  • Theorem 2.5: ${\cal K}$-convergence
  • Remark 3.1