This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

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International audienceIn this paper, we address the problem of solving accurately gas-liquid compressible flows, without pressure oscillations at the gas-liquid interface. We introduce a new Lagrange-projection scheme based on a random sampling technique of Chalons and Goatin. We compare it to a Ghost Fluid approach introduced previously. Despite the non-conservative feature of the schemes, we observe the numerical convergence towards the relevant weak solution, for shock-contact interaction test cases. Finally, we apply the new scheme to the computation of the oscillations of a spherical air bubble inside water

Abstract. In this work we propose an efficient finite volume approximation of two-fluid flows. Our scheme is based on three ingredients. We first construct a conservative scheme that removes the pressure oscillations phenomenon at the interface. The construction relies on a random sampling at the interface [5,6]. Secondly, we replace the exact Riemann solver by a faster relaxation Riemann solver with good stability properties [4]. Finally, we apply Strang directional splitting and optimized memory transpositions in order to achieve high performance on Graphics Processing Unit (GPU) or GPU cluster computations. Résumé. Nous proposons une méthode de volumes finis efficace pour l'approximation d'écoulementsbifluides. Le schémas est basé sur trois ingrédients. Nous construisons d'abord un schéma conservatif sans oscillations de pression. La construction repose sur un échantillonnage aléatoire à l'interface. Ensuite, nous remplaçons le solveur de Riemann exact par un solveur approché de relaxation plus rapide, et qui possède de bonnes propriétés de stabilité. Finalement, nous appliquons un splitting directionnel de Strang et des techniques de transposition optimisées en mémoire pour atteindre de bonnes performances sur GPU ou cluster de GPU.

Abstract. We introduce continuous tools to study the low Mach number behavior of the Godunov scheme applied to the linear wave equation with porosity on cartesian meshes. More precisely, we extend the Hodge decomposition to a weighted L 2 space in the continuous case and we study the properties of the modified equation associated to this Godunov scheme. This allows to partly explain the inaccuracy of the Godunov scheme at low Mach number on cartesian meshes and to propose two corrections: a first one named low Mach and a second one named all Mach. These results are preliminary since it remains to prove them in the discrete case. Résumé. Nous introduisons des outils au niveau continu pourétudier le comportement du schéma deGodunovà bas nombre de Mach appliquéà l'équation des ondes avec porosité sur maillage cartésien. Plus précisément, nousétendons la décomposition de Hodgeà un espace L 2 avec poids et nousétudions les propriétés de l'équation modifiée associée au schéma de Godunov. Cela permet de partiellement expliquer la perte de précision du schéma de Godunovà bas nombre de Mach sur maillage cartésien et de proposer deux corrections: une première dite bas Mach et une seconde dite tout Mach. Ces résultats sont préliminaires puisqu'il resteà lesétendre au niveau discret.

We consider a two-fluid compressible flow. Each fluid obeys a stiffened gas pressure law. The continuous model is well defined without considering mixture regions. However, for numerical applications it is often necessary to consider artificial mixtures, because the two-fluid interface is diffused by the numerical scheme. We show that classic pressure law interpolations lead to a non-convex hyperbolicity domain and failure of well-known numerical schemes. We propose a physically relevant pressure law interpolation construction and show that it leads to a necessary modification of the pure phase pressure laws. We also propose a numerical scheme that permits to approximate the stiffened gas model without artificial mixture.

Classical finite volume schemes for the Euler system  are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

This article is dedicated to the long time behavior of a finite volume approximation of general symmetrizable linear hyperbolic system on a bounded domain. In the continuous case this problem is very difficult, and the $\omega $–limit set (namely the set of all the possible long time limits) may be large and complicated to depict if no dissipation is introduced. In this article we prove that in general, with a stable finite volume scheme, the discrete solution converges to a steady state when the time goes to infinity. This property is a direct consequence of the numerical dissipation mechanisms used for stabilizing the discretization. We apply this result for determining the long time limit for several stabilizations of the wave system, and perform a formal link with the low Mach number problem of the nonlinear Euler system. Numerical experiments with the wave system are performed for confirming the theoretical results obtained.