Florent Berthelin

From Kinetic Equations to Multidimensional Isentropic Gas Dynamics Before Shocks

This article is devoted to the proof of the hydrodynamical limit from kinetic equations (including BGK-like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method; hence the derivation is valid only before shocks appear on the limit system solution. However, no a priori knowledge on high velocity distributions for kinetic functions is needed. The case of the Saint-Venant system with topography (where a source term is added) is included.

EXISTENCE AND WEAK STABILITY FOR A PRESSURELESS MODEL WITH UNILATERAL CONSTRAINT

We consider a two-phase model described by a pressureless gas system with unilateral constraint. We prove weak stability and the existence of weak solutions by passing to the limit in the sticky-blocks dynamics. We obtain the maximum principle on the velocity, the Oleinik entropy condition and local entropy inequalities. Initial data are taken in a very weak sense since the solution can jump initially in time.

Weak solutions for a hyperbolic system with unilateral constraint and mass loss

We consider isentropic gas dynamics equations with unilateral constraint on the density and mass loss. The γ and pressureless pressure laws are considered. We propose an entropy weak formulation of the system that incorporates the constraint and Lagrange multiplier, for which we prove weak stability and existence of solutions. The nonzero pressure model is approximated by a kinetic BGK relaxation model, while the pressureless model is approximated by a sticky-blocks dynamics with mass loss.

Kinetic schemes on staggered grids for barotropic Euler models: entropy-stability analysis

We introduce, in the one-dimensional framework, a new scheme of finite volume type for barotropic Euler equations. The numerical unknowns, namely densities and velocities, are defined on staggered grids. The numerical fluxes are defined by using the framework of kinetic schemes. We can consider general (convex) pressure laws. We justify that the density remains non negative and the total physical entropy does not increase, under suitable stability conditions. Performances of the scheme are illustrated through a set of numerical experiments.

Multifluid Flows: A Kinetic Approach

We propose a numerical scheme for the simulation of fluid–particles flows with two incompressible phases. The numerical strategy is based on a finite volume discretization on staggered grids, with a flavor of kinetic schemes in the definition of the numerical fluxes. We particularly pay attention to the difficulties related to the volume conservation constraint and to the presence of a close-packing term which imposes a threshold on the volume fraction of the disperse phase. We are able to identify stability conditions on the time step to preserve this threshold and the energy dissipation of the original model. The numerical scheme is validated with the simulation of sedimentation flows.

Consistency Analysis of a 1D Finite Volume Scheme for Barotropic Euler Models

This work is concerned with the consistency study of a 1D (staggered kinetic) finite volume scheme for barotropic Euler models. We prove a Lax-Wendroff-like statement: the limit of a converging (and uniformly bounded) sequence of stepwise constant functions defined from the scheme is a weak entropic-solution of the system of conservation laws.

Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws

Summary.We prove the convergence of flux vector splitting schemes associated to hyperbolic systems of conservation laws with a single compatible entropy ηc. We prove estimate on the L2 norm of the gradient of the numerical approximation in the inverse square root of the space increment Δx. This estimate is related to the notion of (strictly) ηc-dissipativity on F+, −F− and Id−λ(F+−F−), where F+, F− is the flux-decomposition. The second tool of the proof is a kinetic formulation of the flux-splitting scheme with three velocities. Then we get a control for all entropies and apply the compensated compactness theory.

Theoretical Study of a Multidimensional Pressureless Model with Unilateral Constraint

The aim of this paper is to extend to multidimensions the study of a pressureless model of a gas system with unilateral constraint. Several difficulties are added with respect to the one-dimensional case. Indeed the geometry of the dynamics of blocks cannot be conserved and to solve this problem, we approximate the motion of each block by discrete jumps in all the directions separately in consecutive time steps. This leads to approximations of solutions for special initial data. Then the stability of these approximations has to be adapted to this new situation. We finally get the existence and the stability of solutions.