Asymptotic-preserving IMEX schemes for the Euler equations of non-ideal gases

TL;DR

The paper develops and analyzes asymptotic-preserving IMEX-RK schemes for the compressible Euler equations with general EOS, including single- and two-scale (acoustic) asymptotics. By coupling these time discretizations with a DG spatial discretization, the authors obtain a high-order method that remains stable and consistent in the low Mach regime without operator splitting or flux relaxation, and works across ideal, SG, and general cubic EOS. They establish AP properties for both single- and two-scale limits and validate the approach through extensive numerical experiments (isentrope vortex, colliding pulses, density layering, open tube flow, Gresho vortex, baroclinic generation), demonstrating correct asymptotic behavior and practical efficiency. The work provides a versatile framework for accurate low-Mach and incompressible flow simulations of non-ideal gases, with implications for atmospheric dynamics, combustion, and high-precision CFD across a broad EOS landscape.

Abstract

We analyze schemes based on a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, a numerical method effective for a wide range of Mach numbers is obtained. A number of benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis.

Figures (16)

• Figure 1: Colliding acoustic pulses test case, pressure profile. Left: $t = \frac{T_{f}}{2}$. Right: $t = T_{f}$. The initial profile is in dashed black line, the solid blue lines provide the results at the corresponding time obtained with the IMEX method at $C_{u} \approx 0.1$, whereas the red dots show the reference results obtained with the explicit method.
• Figure 2: Colliding acoustic pulses test case, pressure profile. Comparison of the IMEX method employing different time step. Left: $t = \frac{T_{f}}{2}$. Right: $t = T_{f}$. The solid blue lines provide the results at the obtained at $C_{u} \approx 0.1$, the black dots show the results obtained at $C_{u} \approx 0.2$, whereas the red crosses report the results obtained at $C_{u} \approx 0.5$.
• Figure 3: Colliding acoustic pulses test case employing the SG-EOS \ref{['eq:sg_eos']} with $p_{\infty} = 6.8 \times 10^{-3}$, pressure profile. Left: $t = \frac{T_{f}}{5}$. Right: $t = T_{f}$. The initial profile is in dashed black line, the solid blue lines provide the results at the corresponding time obtained with the IMEX method, whereas the red dots show the reference results obtained with the explicit method.
• Figure 4: Colliding acoustic pulses test case employing the SG-EOS \ref{['eq:sg_eos']} with $p_{\infty} = 6.8 \times 10^{3}$, pressure profile. Left: $t = \frac{3}{10}T_{f}$. Right: $t = T_{f}$. The solid blue lines provide the results at the corresponding time obtained with the IMEX method at acoustic Courant number $C \approx 80$, the solid black lines report the results obtained with the IMEX method at $C \approx 1$, whereas the red dots show the reference results obtained with the explicit method.
• Figure 5: Density layering test case at $M = 0.02$ with the ideal gas law \ref{['eq:ideal_gas']}. Left: density. Right: pressure. The dashed black lines represent the initial condition, the continuous blue lines show the solution at the final time, whereas the red dots report the solution obtained with the third order optimal explicit SSP scheme.

Theorems & Definitions (5)

• Definition 3.1
• Theorem 3.4
• proof
• Theorem 3.7
• proof