A quantitative comparison of high-order asymptotic-preserving and asymptotically-accurate IMEX methods for the Euler equations with non-ideal gases

TL;DR

This paper compares two high-order IMEX methods for the Euler equations in low Mach regimes, analyzing their asymptotic-preserving and accurate properties for ideal and non-ideal gases using high-order DG discretization.

Contribution

It provides a detailed quantitative comparison of high-order asymptotic-preserving and asymptotically-accurate IMEX methods for non-ideal gas Euler equations, including analysis of nonlinear pressure solutions.

Findings

Both methods effectively handle low Mach flows.

The semi-implicit approach avoids nonlinear pressure equations for non-ideal gases.

High-order accuracy is achieved with DG spatial discretization.

Abstract

We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK approach involves an implicit coupling between the momentum and energy balance so as to avoid the acoustic CFL restriction, while the density can be treated in a fully explicit fashion. This approach leads to a mildly nonlinear equation for the pressure, which can be solved according to a fixed point procedure. An alternative strategy consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases. The spatial discretization is based on a Discontinuous Galerkin (DG) method, which naturally allows high-order accuracy. The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases.