Asymptotic preserving finite volume method for the compressible Euler equations: analysis via dissipative measure-valued solutions

TL;DR

This work develops and analyzes a novel asymptotic preserving finite volume scheme for the multidimensional compressible barotropic Euler equations at low Mach numbers, using a stabilized upwind flux with a velocity shift proportional to the stiff pressure gradient. It employs the framework of dissipative measure-valued (DMV) solutions to rigorously establish AP properties: for fixed mesh, DMV convergence to the compressible Euler system occurs as $\varepsilon$ remains positive, while DMV solutions converge to the incompressible Euler solution as $\varepsilon\to 0$, provided well-prepared data. Conversely, on a fixed mesh, the limit scheme is an energy-stable, consistent approximation of the incompressible Euler equations, with the DMV limit coinciding with the classical solution via weak-strong uniqueness. The paper also proves that the limits commute (AP property) and validates the theory through extensive numerical experiments employing $\mathcal{K}$-convergence and Wasserstein-distance analyses, demonstrating robust convergence behavior across regimes.

Abstract

We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.

Figures (8)

• Figure 1: AP Property of a scheme.
• Figure 2: Dual grid.
• Figure 3: Deviations of density from 1 and momentum profiles when $k=1024$ for $\varepsilon = 1, 10^{-1},\dots, 10^{-4}$.
• Figure 4: Divergence of the velocity field when $k = 1024$ and $\varepsilon = 10^{-4}$ at final time $T = 0.02$.
• Figure 5: $\left\lVert\varrho_\varepsilon(t,\cdot) - 1\right\rVert_{L^\gamma(\Omega)}$ over time for different values of $\varepsilon$ (left). $\left\lVert\varrho_\varepsilon - 1\right\rVert_{L^\infty(0,T;L^\gamma(\Omega))}$ versus $\varepsilon$ (right). Results are computed on a $128\times 128$ grid.

Theorems & Definitions (72)

• Proposition 2.1
• Definition 2.2: Dissipative Measure-Valued Solution
• Remark 2.3
• Remark 2.4
• Theorem 2.5: Weak-Strong Uniqueness Principle
• Lemma 2.6: Energy Estimates for Weak and DMV Solutions
• proof
• Remark 2.7
• Proposition 2.8: A priori energy estimates for the modified system
• Theorem 3.1