Simulations of blood flows in arteries require numerical solutions of fluidstructure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels.Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one that involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem.

Abstract. FreeFem ++ [11] is a software for the numerical solution of partial dierential equations. It is based on nite element method. The FreeFem ++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem ++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project.First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed.

Abstract. In Inertial Confinement Fusion (ICF) simulation, use of Lagrangian hydrodynamic numerical schemes is a cornerstone. It avoids mixing of materials and allows for symmetry preservation in dimension two. Recently, [7] and then [9] proposed an interesting alternative to the historical VNR scheme [15]. These two first order schemes are multidimensional generalizations of the Godunov acoustic solver. Alternatively, a WENO Lagrangian scheme was proposed in [6]. This scheme suffers from non-preservation of symmetries and its velocity computation can be discussed.The aim of this work is to evaluate the later scheme on ICF representative test cases and to derive a polynomial reconstruction that preserves symmetries for the three cell-centered scheme. This reconstruction is inspired by [12]. Since this paper focuses on the approximation of Euler equations, considered test cases are purely hydrodynamic and do not illustrate all difficulties encountered in ICF.We first briefly recall different schemes used for this study. We then explain the Least-Squares ENO reconstruction that we chose for symmetry preservation and describe the limiting strategy. We finally illustrates the presented results by some representative numerical experiments.Résumé. La simulation de Fusion par Confinement Inertiel (FCI) utilise souvent des schémas hydrodynamiques Lagrangiens. Cela permet d'éviter le mélange de matériaux et permet de préserver des symétries en dimension deux. Récemment, des alternatives intéressantes au schéma historique VNR [15] ontété proposées dans [7] puis dans [9]. Parallèlement, un schéma WENO aété proposé dans [6]. Ce schéma ne préserve pas les symétries et le calcul des vitesses peutêtre discuté. L'objectif de ce travail est d'évaluer ce dernier schéma pour des cas tests représentatifs de la FCI et d'écrire une reconstruction polynomiale qui préserve les symétries pour les trois schémasétudiés. Cette reconstruction est inspirée de [12]. Puisqu'on se concentre ici sur la résolution deséquations d'Euler, les cas tests présentés sont purement hydrodynamiques et ne prétendent pas couvrir l'ensemble des difficultés rencontrées en FCI.Nous rappelons d'abord les différents schémas utilisés ici. Nous expliquons ensuite la reconstruction au sens des moindres carrés que nous avons choisie ainsi qu'une stratégie de limitation préservant aussi la symétrie. On illustre finalement les résultats au travers de quelques cas tests représentatifs choisis.

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Abstract. This work aims to extend in two distinct directions results recently obtained in [10]. In a first step we focus on the possible extension of our results to the time dependent case. Whereas in the second part some preliminary numerical simulations aim to give orders of magnitudes in terms of numerical costs of direct 3D simulations.We consider, in the first part, the time dependent rough problem for a simplified heat equation in a straight channel that mimics the axial velocity under an oscillating pressure gradient. We derive first order approximations with respect to , the size of the roughness. In order to understand the problem and set up correct boundary layer approximations, we perform a time periodic fourier analysis and check that no frequency can interact with the roughness. We show rigorously on this toy problem that the boundary layers remain stationary in time (independent on the frequency number). Finally we perform numerical tests validating our theoretical approach.In the second part, we determine actual limits, when running three-dimensional blood flow simulations of the non-homogenized stented arteries. We solve the stationary Stokes equations for an artery containing a saccular aneurysm. Consecutive levels of uniform mesh refinement, serve to relate spatial resolution, problem scale, and required computation time. Test computations are presented for femoral side aneurysm, where a simplified ten-wire stent model was placed across the aneurysm throat. We advocate the proposed stent homogenization model, by concluding that an actual computation power is not sufficient to run accurate, direct simulations of a pulsatile flow in stented vessels.