Fabrice Deluzet

Pierre Degond, Fabrice Deluzet, Alexei Lozinski et al.

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

Stéphane Brull, Pierre Degond, Fabrice Deluzet

This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem. The numerical method is designed for arbitrary space-dependent anisotropy directions and does not require any specially adapted coordinate system. It is also designed to be equally accurate in the strongly and the mildly anisotropic cases. The method is applied to the Euler-Lorentz system, in the drift-fluid limit. This system provides a model for magnetized plasmas.

Stéphane Brull, Fabrice Deluzet, Alexandre Mouton

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter \varepsilon, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit \varepsilon \infty 0 is demonstrated to be a singular perturbation of the original diffusion equation. To address efficiently this problem, an Asymptotic-Preserving scheme is derived. This numerical method does not require the use of coordinates adapted to the anisotropy direction and exhibits an accuracy as well as a computational cost independent of the anisotropy strength.

Anais Crestetto, Fabrice Deluzet, Claudia Negulescu

This paper presents a hybrid numerical method to solve efficiently a class of highly anisotropic elliptic problems. The anisotropy is aligned with one coordinate-axis and its strength is described by a parameter $\eps \in (0,1]$, which can largely vary in the study domain. Our hybrid model is based on asymptotic techniques and couples (spatially) an Asymptotic-Preserving model with its asymptotic Limit model, the latter being used in regions where the anisotropy parameter $\eps$ is small. Adequate coupling conditions link the two models. Aim of this hybrid procedure is to reduce the computational time for problems where the region of small $\eps$-values extends over a significant part of the domain, and this due to the reduced complexity of the limit model.

Pierre Degond, Fabrice Deluzet, David Doyen

In this article, we design Asymptotic-Preserving Particle-In-Cell methods for the Vlasov-Maxwell system in the quasi-neutral limit, this limit being characterized by a Debye length negligible compared to the space scale of the problem. These methods are consistent discretizations of the Vlasov-Maxwell system which, in the quasi-neutral limit, remain stable and are consistent with a quasi-neutral model (in this quasi-neutral model, the electric field is computed by means of a generalized Ohm law). The derivation of Asymptotic-Preserving methods is not straightforward since the quasi-neutral model is a singular limit of the Vlasov-Maxwell model. The key step is a reformulation of the Vlasov-Maxwell system which unifies the two models in a single set of equations with a smooth transition from one to another. As demonstrated in various and demanding numerical simulations, the Asymptotic-Preserving methods are able to treat efficiently both quasi-neutral plasmas and non-neutral plasmas, making them particularly well suited for complex problems involving dense plasmas with localized non-neutral regions.

Pierre Degond, Fabrice Deluzet, Claudia Negulescu

In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of the elliptic operator being supplemented with Neumann boundary conditions. A new scheme is introduced which allows an accurate resolution of this elliptic equation for an arbitrary anisotropy ratio.