Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system

International audience; This article is composed of three self-consistent chapters that can be read independently of each other. In Chapter 1, we define and we 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 Chapter 2, we apply the all Mach correction proposed in Chapter 1 to the case of the non-linear 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 non-linear case by showing how this construction is related with the linear analysis of Chapter 1. At last, we apply in Chapter 3 the all Mach correction proposed in Chapter 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Chapters 1, 2 and 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.