Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

Abstract

This article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency and they suffer of severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic waves speed. In this work, we propose and analyze a new unconditionally stable an consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. A stability analysis and several one and two dimensional simulations confirm that the proposed method possesses the sought characteristics. 1. Introduction. Almost all fluids can be said to be compressible. However, there are many situations in which the changes in density are so small to be considered negligible. We refer to these situations saying that the fluid is in an incompressible regime. From the mathematical point of view, the difference between compressible and incompressible situations is that, in the second case, the equation for the conservation of mass is replaced by the constraint that the divergence of the velocity should be zero. This is due to the fact that when the Mach number tends to zero, the pressure waves can be considered to travel at infinite speed. From the theoretical point of view, researchers try to fill the gap between those two different descriptions by determining in which sense compressible equations tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in the numerical solution of the Euler system when used to describe fluid flows where the Mach number strongly varies. This causes the gas to pass from compressible to almost incompressible situations and consequently it causes most of the numerical methods build for solving compressible Euler equations to fail. In fact, when the Mach number tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in the incompressible limit. This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods [34] or by splitting and correcting the pressure on the collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see for instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of the acoustic waves in the fluid in order to remain stable. This means that they suffer from a restrictive CFL (Courant-Frierichs-Levy) condition which is inversely proportional to the Mach number value. In this work, we derive a method which