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Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations

Megala Anandan, Mária Lukáčová-Medvid'ová

Abstract

We develop structure-preserving finite volume schemes for the barotropic Euler equations in the low Mach number regime. Our primary focus lies in ensuring both the asymptotic-preserving (AP) property and the discrete entropy stability. We construct an implicit-explicit (IMEX) method with suitable acoustic/advection splitting including implicit numerical diffusion that is independent of the Mach number. We prove the positivity of density, the entropy stability, and the asymptotic consistency of the fully discrete numerical method rigorously. Numerical experiments for benchmark problems validate the structure-preserving properties of the proposed method.

Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations

Abstract

We develop structure-preserving finite volume schemes for the barotropic Euler equations in the low Mach number regime. Our primary focus lies in ensuring both the asymptotic-preserving (AP) property and the discrete entropy stability. We construct an implicit-explicit (IMEX) method with suitable acoustic/advection splitting including implicit numerical diffusion that is independent of the Mach number. We prove the positivity of density, the entropy stability, and the asymptotic consistency of the fully discrete numerical method rigorously. Numerical experiments for benchmark problems validate the structure-preserving properties of the proposed method.

Paper Structure

This paper contains 26 sections, 16 theorems, 82 equations, 8 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Main contributions
  4. Outline
  5. Preliminaries
  6. Space discretization
  7. Discrete function spaces
  8. Discrete difference operators
  9. Numerical scheme
  10. Positivity of density
  11. Discrete energy inequality property
  12. Asymptotic-preserving property
  13. Implementation of the scheme
  14. Asymptotic consistency
  15. Numerical validation
  16. ...and 11 more sections

Key Result

Proposition 1.1

Suppose that the initial data for the compressible system Euler mass-Euler mom satisfies the regularity $(\varrho^0(\cdot),\mathbf{u}^0(\cdot)) \in H^s(\Omega;\mathbb{R}^{d+1})$ with $s>\frac{d}{2}+2$, and is well-prepared, i.e. such that $\varrho_0^0$ is constant and $\text{div}_x u_0^0 = \text{div}_x u_1^0 = 0$. Then, the limit of the system ND Euler mass-ND Euler mom as $\varepsilon \to 0$ is

Figures (8)

  • Figure 1: Example \ref{['Subsec: SPP']} - Standard periodic problem: Top - Total energy; Middle - Kinetic energy; Bottom - Potential energy
  • Figure 2: Example \ref{['Subsec:CAW']} - Colliding acoustic waves problem: Top: Density at t=0.04,0.06,0.08 (left to right); Middle - velocity at t=0.04,0.06,0.08 (left to right); Bottom - Total energy, kinetic energy, potential energy (left to right)
  • Figure 3: Example \ref{['Subsec:RP']} - Riemann problem: Top to Bottom: Density, velocity, total energy, kinetic energy, potential energy
  • Figure 4: Example \ref{['Subsec:GV']} - Gresho vortex problem: Top - Total energy; Middle - Kinetic energy; Bottom - Potential energy ($\lambda=1$)
  • Figure 5: Example \ref{['Subsec:GV']} - Gresho vortex problem: Surface plots of $\text{div}_h \mathbf{u}_h$ ($\lambda=1$)
  • ...and 3 more figures

Theorems & Definitions (31)

  • Proposition 1.1: Asymptotic limit (see Klainerman and Majda ap_majda1)
  • Theorem A: Positivity of density
  • Theorem B: Discrete total energy inequality
  • Theorem C: Asymptotic-preserving property
  • Lemma 2.1: Discrete grad-div duality
  • Lemma 2.2: Discrete inequalities
  • Lemma 3.1: Discrete renormalization identity
  • proof
  • Theorem 3.1: Positivity of density
  • proof
  • ...and 21 more