In this paper we present and implement the Palindromic Discontinuous Galerkin (PDG) method in dimensions higher than one. The method has already been exposed and tested in [4] in the one-dimensional context. The PDG method is a general implicit high order method for approximating systems of conservation laws. It relies on a kinetic interpretation of the conservation laws containing stiff relaxation terms. The kinetic system is approximated with an asymptotic-preserving high order DG method. We describe the parallel implementation of the method, based on the StarPU runtime library. Then we apply it on preliminary test cases.

In this note, we introduce a new finite volume scheme for Euler equations with source terms: the gravity and the friction. The classical finite volume schemes are not able to capture correctly the dynamic induced by the balance between convective terms and external forces. Firstly, by plugging the source terms in the fluxes with the Jin-Levermore procedure, we modify the Lagrangian+remap scheme to obtain a method able to capture the asymptotic limit induced by the friction (asymptotic preserving scheme) and discretize with good accuracy the steady state linked to the gravity (well-balanced scheme). Secondly we will give some properties about this scheme and introduce a modification which allows us to obtain an arbitrary high order discretization of the hydrostatic steady state.

Abstract. This paper introduces a Semi-Lagrangian solver for the Vlasov-Poisson equations on a uniform hexagonal mesh. The latter is composed of equilateral triangles, thus it doesn't contain any singularities, unlike polar meshes. We focus on the guiding-center model, for which we need to develop a Poisson solver for the hexagonal mesh in addition to the Vlasov solver. For the interpolation step of the Semi-Lagrangian scheme, a comparison is made between the use of Box-splines and of Hermite finite elements. The code will be adapted to more complex models and geometries in the future. Résumé. Dans cet article nous présentons un solveur semi-Lagrangien pour leséquations de Vlasov-Poisson sur un maillage hexagonal uniforme. Ce dernier est composé de triangleséquilatéraux, ainsi il ne présente aucune singularité, contrairement au maillage polaire. Nous nous concentrons ici sur le modèle centre-guide.À cette fin nous avons développé en plus du solveur pour Vlasov, un solveur de l'équation de Poisson pour maillage hexagonal. Nous comparons les résultats obtenus avec une interpolation paréléments finis d'Hermite et par des Box-splines. Dans l'avenir, ce code sera adaptéà des géométries et modèles plus complexes.