Mathematical and numerical approaches for multiscale problems
Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem
We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be –periodic, being the magnitude of the mean thickness of the layer, and a positive parameter describing the degree of roughness. For tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent...
A brief introduction to homogenization and miscellaneous applications
This paper is a set of lecture notes for a short introductory course on homogenization. It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the oscillating test function method and two-scale convergence) and briefly describes the main results of the more general theory of − or −convergence. Several applications of the method are given: derivation of Darcy’s law for flows in porous media, derivation of the porosity...
Numerical homogenization: survey, new results, and perspectives
These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the...
Numerical study of acoustic multiperforated plates
It is rather classical to model multiperforated plates by approximate impedance boundary conditions. In this article we would like to compare an instance of such boundary conditions obtained through a matched asymptotic expansions technique to direct numerical computations based on a boundary element formulation in the case of linear acoustic.
Numerical simulations of the focal spot generated by a set of laser beams : LMJ
In order to get the fusion of small capsules containing a deuterium-tritium nuclear fuel, the MegaJoule laser (LMJ) will focus a large number of laser beams inside a cylinder (Hohlraum) which contains the fusion capsule. In order to control this process we have to know as well as possible the electromagnetic field created by the laser beams on both Hohlraum’s apertures. This article describes a numerical tool which computes this electromagnetic field...
Wall laws for viscous fluids near rough surfaces
In this paper, we review recent results on wall laws for viscous fluids near rough
surfaces, of small amplitude and wavelength
Asymptotic-Preserving scheme for a two-fluid Euler-Lorentz model
The present work is devoted to the simulation of a strongly magnetized plasma as a mixture of an ion fluid and an electron fluid. For simplicity reasons, we assume that each fluid is isothermal and is modelized by Euler equations coupled with a term representing the Lorentz force, and we assume that both Euler systems are coupled through a quasi-neutrality constraint of the form = . The numerical method...
Directional and scale-dependent statistics of quasi-static magnetohydrodynamic turbulence
Anisotropy and intermittency of quasi-static magnetohydrodynamic (MHD) turbulence in an imposed magnetic field are examined, using three-dimensional orthonormal wavelet analysis. Wavelets are an efficient tool to examine directional scale-dependent statistics, since they are based on well-localized functions in space, scale and direction. The analysis is applied to two turbulent MHD flows computed by direct numerical simulation with 512 grid points...
An axisymmetric PIC code based on isogeometric analysis
Isogeometric analysis has been developed recently to use basis functions resulting from the CAO description of the computational domain for the finite element spaces. The goal of this study is to develop an axisymmetric Finite Element PIC code in which specific spline Finite Elements are used to solve the Maxwell equations and the same spline functions serve as shape function for the particles. The computational domain itself is defined using splines...
Foreword of the CEMRACS’10 research projects publications
Conservative numerical methods for a two-temperature resistive MHD model with self-generated magnetic field term
We propose numerical methods on Cartesian meshes for solving the 2-D axisymmetric two-temperature resistivive magnetohydrodynamics equations with self-generated magnetic field and Braginskii’s [1] closures. These rely on a splitting of the complete system in several subsystems according to the nature of the underlying mathematical operator. The hyperbolic part is solved using conservative high-order dimensionally split Lagrange-remap schemes whereas...
Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson
We present a discontinuous Galerkin scheme for the numerical approximation of the one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.
An asymptotic preserving scheme for model using classical diffusion schemes on unstructured polygonal meshes
A new scheme for discretizing the model on unstructured polygonal meshes is proposed. This scheme is designed such that its limit in the diffusion regime is the MPFA-O scheme which is proved to be a consistent variant of the Breil-Maire diffusion scheme. Numerical tests compare this scheme with a derived GLACE scheme for the system.
Extension of ALE methodology to unstructured conical meshes
We propose a bi-dimensional finite volume extension of a continuous ALE method on unstructured cells whose edges are parameterized by rational quadratic Bezier curves. For each edge, the control point possess a weight that permits to represent any conic (see for example [LIGACH]) and thanks to [WAGUSEDE,WAGU], we are able to compute the of our cells. We then give an extension of scheme for remapping step based on volume fluxing [MARSHA] and self-intersection...
A second order anti-diffusive Lagrange-remap scheme for two-component flows
We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [1]. The algorithm is based on [8] which deals with a non-dissipative first order resolution in Lagrange-remap formalism. In the present paper we describe, in the same framework, an algorithm that is second order accurate in time and space, and that...
Numerical approximation of Knudsen layer for the Euler-Poisson system
In this work, we consider the computation of the boundary conditions for the linearized Euler–Poisson derived from the BGK kinetic model in the small mean free path regime. Boundary layers are generated from the fact that the incoming kinetic flux might be far from the thermodynamical equilibrium. In [2], the authors propose a method to compute numerically the boundary conditions in the hydrodynamic limit relying on an analysis of the boundary layers....
Solving the Vlasov equation in complex geometries
This paper introduces an isoparametric analysis to solve the Vlasov equation with a semi-Lagrangian scheme. A Vlasov-Poisson problem modeling a heavy ion beam in an axisymmetric configuration is considered. Numerical experiments are conducted on computational meshes targeting different geometries. The impact of the computational grid on the accuracy and the computational cost are shown. The use of analytical mapping or Bézier patches does not induce...
Finite volume method in curvilinear coordinates for hyperbolic conservation laws
This paper deals with the design of finite volume approximation of hyperbolic conservation laws in curvilinear coordinates. Such coordinates are encountered naturally in many problems as for instance in the analysis of a large number of models coming from magnetic confinement fusion in tokamaks. In this paper we derive a new finite volume method for hyperbolic conservation laws in curvilinear coordinates. The method is first described in a general...
Hybrid model for the Coupling of an Asymptotic Preserving scheme with the Asymptotic Limit model: The One Dimensional Case
In this paper a strategy is investigated for the spatial coupling of an asymptotic preserving scheme with the asymptotic limit model, associated to a singularly perturbed, highly anisotropic, elliptic problem. This coupling strategy appears to be very advantageous as compared with the numerical discretization of the initial singular perturbation model or the purely asymptotic preserving scheme introduced in previous works [3, 5]. The model problem addressed...
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