A Low Cost Semi-implicit Low-Mach Relaxation Scheme for the Full Euler Equations

Abstract

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut, Chalons, Guisset (2019) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves.

Semi-implicit Two-Speed Well-Balanced Relaxation Scheme for Ripa Model

“…However the scheme is not able to treat wet/dry transitions. The 2D extension have been performed for the Euler system in [5]. In the future, we propose to extend the method to flows around equilibria MHD system.…”

“…The proof is based on a classical Chapman-Enskog expansion detailed for the Euler system case in [5]. Note that an equilibrium given by ∂ x p = −ghΘ∂ x z and u = 0 is still a steady state of the first order approximation (4) of the relaxation system.…”

“…This property is necessary, but not sufficient, to obtain a well-balanced scheme since we will discretize the relaxation system (3) and not the original one. In [5], the entropy stability of (4) is analysed. This computation adapted to our model gives the following stability conditions: β ≥ 0, γ ≥ 0.…”

“…To discretize two-scale hyperbolic problems with a good accuracy, several methods have been proposed and are generally based on semi-implicit schemes to prevent us from the fast scale stringent stability condition. In [3,5], such a scheme have been proposed and it is based on very recent relaxation method and a dynamical splitting. This method allows to adapt the time discretization to the regime and the implicit step just reduces to the inversion of a discrete Laplacian with constant coefficient.…”

A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity

“…The key advantage of this method is that under a subcharacteristic condition it is stable and provably entropy satisfying. Later the two-speed relaxation system was used to develop an IMEX scheme for the homogeneous Euler equations [9]. In contrast, the method presented in this paper is fully explicit.…”