Emmanuel Franck

Asymptotic Preserving Finite Volumes Discretization For Non-Linear Moment Model On Unstructured Meshes

In this work we present a new finite volume discretization of the nonlinear model M 1 [2]. This new method is based on nodal solver for hyperbolic systems [3, 6] and overcomes, on 2-D unstructured meshes, the problem of the inconsistent diffusion limit for schemes based on classical edge formulation. We provide numerical examples to illustrate the properties of the method.

Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes

We prove the uniform AP convergence on unstructured meshes in 2D of a generalization, of the Gosse-Toscani 1D scheme for the hyperbolic heat equation. This scheme is also a nodal extension in 2D of the Jin-Levermore scheme described in [18] for the 1D case. In 2D, the proof is performed using a new diffusion scheme.

Modified Finite Volume Nodal Scheme for Euler Equations with Gravity and Friction

In this work we present a new finite volume scheme valid on unstructured meshes for the Euler equation with gravity and friction indeed the classical Godunov type schemes are not adapted to treat the hyperbolic systems with source terms. The new method is based on a finite volume nodal scheme modified to capture correctly the behavior induced by the source terms.

Finite Volume Scheme with Local High Order Discretization of the Hydrostatic Equilibrium for the Euler Equations with External Forces

A new finite volume scheme for the Euler equations with gravity and friction source terms is presented. Classical finite volume schemes are not able to capture correctly the dynamics generated by the balance between convective terms and external forces. Our purpose is to develop a method better suited for dealing with this problem. To that end, firstly, we modify the Lagrangian+remap scheme by plugging the source terms into the fluxes using the Jin–Levermore procedure. The scheme obtained is able to capture the asymptotic limit induced by the friction (Asymptotic Preserving scheme) and to discretize with a good accuracy the steady-state linked to gravity (Well-Balanced scheme). Secondly, we present some properties about this scheme and introduce a modification for an arbitrary high order discretization of the hydrostatic steady-state.

Palindromic Discontinuous Galerkin Method

We present a high-order scheme for approximating kinetic equations with stiff relaxation. The construction is based on a high-order, implicit, upwind Discontinuous Galerkin formulation of the transport equations. In practice, because of the triangular structure of the implicit system, the computations are explicit. High order in time is achieved thanks to a palindromic composition method. The whole method is asymptotic-preserving with respect to the stiff relaxation and remains stable even with large CFL numbers.