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

TL;DR

The paper quantitatively compares asymptotic-preserving IMEX-RK methods for the Euler equations in the low Mach limit, contrasting a standard IMEX-DG scheme with implicit pressure coupling against a semi-implicit SI-IMEX-DG variant that linearizes EOS effects to avoid nonlinear pressure solves. Both methods achieve AP and AA properties, maintaining high-order accuracy in space via DG and delivering robust performance across ideal and non-ideal EOS, including Peng–Robinson. A detailed treatment of EOS impact, fixed-point strategies, and flux choices demonstrates substantial time-step advantages for SI-IMEX, with type I schemes offering superior low-Mach stability. The results, across Taylor–Green and traveling vortices, open-tube flows, shock-tube, and KH-instability benchmarks, show that SI-IMEX can deliver equivalent accuracy with notable computational savings, even in non-ideal EOS scenarios. The study also outlines practical considerations for spatial discretization and points to future work on simplicial/Voronoi meshes and compatible finite elements to further improve low-Mach scaling and extend to more complex flows.

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.

Figures (13)

• Figure 1: Example of two neighboring elements for a nodal DG formulation based on Lagrange polynomials. The nodes correspond to the support points of $\left(r + 1\right)$-order Gauss-Lobatto quadrature rule (in the image $r= 1$).
• Figure 2: Taylor-Green vortex test case, time evolution of the divergence of the velocity at $M = 10^{-3}$. The results are obtained using the IMEX method with the IMEX-RK(3,3,3) scheme of type II in Table \ref{['tab:rk3_butch_type_II']} together with polynomial degree $r = 2$ and $N_{el} = 60$.
• Figure 3: Traveling vortex test case with the IMEX-RK(2,2,2) scheme (Table \ref{['tab:rk2_butch']}) and polynomial degree $r = 1$, contour plots of the pressure perturbation $p - (1 - M^{2})$ at $M = 10^{-2}$ with $N_{el} = 160$. Left: initial field. Right: field at final time $t = T_{f} = 3$.
• Figure 4: Traveling vortex test case with the IMEX-RK(2,2,2) scheme (Table \ref{['tab:rk2_butch']}), $r = 2$ for the velocity field and $r = 1$ for the remaining variables, contour plots of the pressure perturbation $p - (1 + M^{2})$ at $M = 10^{-2}$ with $N_{el} = 160$. Left: initial field. Right: field at final time $t = T_{f} = 3$.
• Figure 5: Traveling vortex test case, contour plot of the velocity perturbation at $M = 10^{-4}$. Left: initial field. Right: field at final time $t = T_{f} = 3$. The results are obtained using the IMEX method with the IMEX-RK(3,3,3) scheme (Table \ref{['tab:rk3_butch_type_II']}) and polynomial degree $r = 2$ with $N_{el} = 40$.

Theorems & Definitions (2)

• Definition 3.1
• Definition 3.2