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Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations

Megala Anandan, K. R. Arun, Amogh Krishnamurthy, Mária Lukáčová-Medvid'ová

Abstract

We design an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number $\varepsilon>0$, we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in $\varepsilon$ and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as $\varepsilon$, and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.

Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations

Abstract

We design an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number , we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as , and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.

Paper Structure

This paper contains 25 sections, 15 theorems, 123 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. AP schemes
  4. Relative energy
  5. Main contributions
  6. Outline
  7. Compressible Euler System and its Strong Solutions
  8. Singular limit of the Euler system
  9. Finite Volume Scheme
  10. Mesh and Unknowns
  11. Numerical Scheme
  12. Energy Stability
  13. Consistency
  14. Error Estimates
  15. Error Estimates in the Compressible Regime
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with a smooth boundary $\partial \Omega$ and let $\varepsilon>0$ be fixed. Suppose that the initial data ($\varrho_0, {\bold{u}}_0$), with $\varrho_0>0$, belong to $H^{s}(\Omega;\mathbb{R}^{d+1})$ with $s>\frac{d}{2}+1$. Then, there exists $T_{ma

Figures (4)

  • Figure 1: Dual grid.
  • Figure 2: The deviation of the density $\varrho$ from 1 at the final time $T = 0.1$ for different values of $\varepsilon$ on a $512\times 512$ grid.
  • Figure 3: The flow Mach number at the final time $T = 0.1$ for different values of $\varepsilon$ on a $512\times 512$ grid.
  • Figure 4: Relative kinetic energy over time for different values of $\varepsilon$.

Theorems & Definitions (24)

  • Theorem 2.1: Existence of Strong Solutions
  • Corollary 2.2
  • proof
  • Definition 3.1: Well-Prepared Initial Data
  • Theorem 3.2: Existence of Strong Solutions (Incompressible)
  • proof
  • Corollary 3.3
  • proof
  • Theorem 4.1: Energy Stability
  • Theorem 4.2: Existence and Positivity
  • ...and 14 more