Frédéric Lagoutière

Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics

We present a non-diffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the Ultra-Bee limiter of [24], [29]. We prove for the Ultra-Bee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e., piecewise constant) initial data, and state a conjecture of non-diffusion at infinite time, based on some local over-compressivity of the scheme, for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.

An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model

We propose a discretization method of a five-equation model with isobaric closure for the simulation of interfaces between compressible fluids. This numerical solver is a Lagrange-Remap scheme that aims at controlling the numerical diffusion of the interface between both fluids. This method does not involve any interface reconstruction procedure. The solver is equipped with built-in stability and consistency properties and is conservative with respect to mass, momentum, total energy and partial masses. This numerical scheme works with a very broad range of equations of state, including tabulated laws. Properties that ensure a good treatment of the Riemann invariants across the interface are proven. As a consequence, the numerical method does not create spurious pressure oscillations at the interface. We show one-dimensional and two-dimensional classic numerical tests. The results are compared with the approximate solutions obtained with the classic upwind Lagrange-Remap approach, and with experimental and previously published results of a reference test case.

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.

Coupling of general Lagrangian systems

This work is devoted to the coupling of two fluid models, such a s two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is characteristic, a particula r choice of dependent variables which are transmitted ensures the overall conservativity.

Numerical resolution of a two-component compressible fluid model with interfaces

We study an algorithm for moving interfaces in a two-component compressible fluid model. We propose to use the limited downwind scheme developed in Despres and Lagoutiere (1999, 2002) to avoid artificial numerical spreading of interfaces. The numerical treatment of the mixture is shown to be free of spurious pressure and velocity oscillations near the contact discontinuity. The algorithm is conservative for partial masses, total impulse and total energy. Various numerical simulations show the interest of this approach, for interfaces in dimensions 1, 2 and 3. Simplicity of the coding is an important feature of the algorithm.

Probabilistic Analysis of the Upwind Scheme for Transport Equations

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the scheme is of order 1/2 in

Liquid jet generation and break-up

{L^{\infty}([0,T],L^1(\mathbb R^d))}

Small solids in an inviscid fluid

for an integrable initial datum of bounded variation and of order 1/2−ε, for all ε > 0, in