A Godunov-type method for the seven-equation model of compressible two-phase flow

We are interested in the numerical approximation of the solutions of the compressible seven-equation two-phase flow model. We propose a numerical srategy based on the derivation of a simple, accurate and explicit approximate Riemann solver. The source terms associated with the external forces and the drag force are included in the definition of the Riemann problem, and thus receive an upwind treatment. The objective is to try to preserve, at the numerical level, the asymptotic property of the solutions of the model to behave like the solutions of a drift-flux model with an algebraic closure law when the source terms are stiff. Numerical simulations and comparisons with other strategies are proposed

Automatic coupling and finite element discretization of the Navier-Stokes and heat equations

We consider the finite element discretization of the Navier-Stokes equations coupled with the heat equation where the viscosity depends on the temperature. We prove a posteriori error estimates which allow us to automatically determine the zone where the temperature-dependent viscosity must be inserted into the Navier-Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations

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

International audienceThis 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

Godunov-type schemes for hyperbolic systems with parameter dependent source. The case of Euler system with friction

Well balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties

Characterization of the Singular Part of the Solution of Maxwell's Equations in a Polyhedral Domain

The solution of Maxwell's equations in a non-convex polyhedral domain is less regular than in a smooth or convex polyhedral domain. In this paper we show that this solution can be decomposed into the orthogonal sum of a singular part and a regular part, and we give a characterization of the singular part. We also notice that the decomposition is linked to the one associated to the scalar Laplacian

ETUDE MATHEMATIQUE ET NUMERIQUE DE STABILITE POUR DES MODELES HYDRODYNAMIQUES AVEC TRANSITION DE PHASE

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