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An asymptotic preserving scheme satisfying entropy stability for the barotropic Euler system

Megala Anandan, Mária Lukáčová-Medvid'ová, S. V. Raghurama Rao

TL;DR

The paper tackles the challenge of simulating the barotropic Euler equations in the low Mach regime by developing asymptotic preserving (AP) numerical schemes that also satisfy discrete entropy stability. It combines an implicit-explicit (IMEX) time discretisation with three space discretisation strategies, including an entropy-stable flux, to maintain stability across all ε. A formal entropy inequality depending on ε is derived, and AP properties are established both for the time semi-discrete and fully discrete schemes. Numerical experiments across standard low-Mach tests (standard periodic, colliding waves, Riemann, Gresho, travelling vortex) demonstrate robust entropy decay and convergence that remains accurate as ε → 0, highlighting the approach’s practical relevance for compressible-incompressible transitions.

Abstract

In this paper we study structure-preserving numerical methods for low Mach number barotropic Euler equations. Besides their asymptotic preserving properties that are crucial in order to obtain uniformly consistent and stable approximations of the Euler equations in their singular limit as the Mach number approaches zero, our aim is also to preserve discrete entropy stability. Suitable acoustic/advection splitting approach combined with time implicit-explicit approximations are used to achieve the asymptotic preserving property. The entropy stability of different space discretisation strategies is studied for different values of Mach number and is validated by the numerical experiments.

An asymptotic preserving scheme satisfying entropy stability for the barotropic Euler system

TL;DR

The paper tackles the challenge of simulating the barotropic Euler equations in the low Mach regime by developing asymptotic preserving (AP) numerical schemes that also satisfy discrete entropy stability. It combines an implicit-explicit (IMEX) time discretisation with three space discretisation strategies, including an entropy-stable flux, to maintain stability across all ε. A formal entropy inequality depending on ε is derived, and AP properties are established both for the time semi-discrete and fully discrete schemes. Numerical experiments across standard low-Mach tests (standard periodic, colliding waves, Riemann, Gresho, travelling vortex) demonstrate robust entropy decay and convergence that remains accurate as ε → 0, highlighting the approach’s practical relevance for compressible-incompressible transitions.

Abstract

In this paper we study structure-preserving numerical methods for low Mach number barotropic Euler equations. Besides their asymptotic preserving properties that are crucial in order to obtain uniformly consistent and stable approximations of the Euler equations in their singular limit as the Mach number approaches zero, our aim is also to preserve discrete entropy stability. Suitable acoustic/advection splitting approach combined with time implicit-explicit approximations are used to achieve the asymptotic preserving property. The entropy stability of different space discretisation strategies is studied for different values of Mach number and is validated by the numerical experiments.

Paper Structure

This paper contains 23 sections, 2 theorems, 64 equations, 13 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. The barotropic Euler system
  4. Entropy stability property
  5. Asymptotic limit
  6. Numerical method
  7. Semi-discrete IMEX time discretisation
  8. Asymptotic preserving property of the time semi-discrete scheme
  9. Space discretisation
  10. Type 1
  11. Type 2
  12. Type 3
  13. Asymptotic preserving property of the fully discrete scheme
  14. Numerical results
  15. Standard periodic problem
  16. ...and 8 more sections

Key Result

Theorem 1

Assume well-prepared initial conditions in wp rho ic and wp u ic, the asymptotic expansion in AP ansatz 1-AP ansatz 3, and periodic boundary conditions on $\rho$ and $\mathbf{u}$. Then the time semi-discrete GSA IMEX-RK scheme given by sd stage time_mass2 and sd stage time_mom satisfies for $\epsilo for all $i \in \{1,2,\dots,s \}$, which is a consistent approximation of the incompressible Euler s

Figures (13)

  • Figure 1: Standard periodic problem: Entropy, KE and PE plots for $\epsilon=0.5$ using space discretisation types 1, 2 and 3.
  • Figure 2: Standard periodic problem: Entropy, KE and PE plots for $\epsilon=0.1$ using space discretisation types 1, 2 and 3.
  • Figure 3: Standard periodic problem: Entropy, KE and PE plots for $\epsilon=0.0001$ using space discretisation types 1, 2 and 3.
  • Figure 4: Colliding acoustic waves problem with $\epsilon=0.1$ using type 1 space discretisation.
  • Figure 5: Colliding acoustic waves problem with $\epsilon=0.1$ using type 2 space discretisation.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof