Claudia Negulescu

Pierre Degond, Alexei Lozinski, Jacek Narski et al.

The concern of the present work is the introduction of a very efficient Asymptotic Preserving scheme for the resolution of highly anisotropic diffusion equations. The characteristic features of this scheme are the uniform convergence with respect to the anisotropy parameter $0<\eps <<1$, the applicability (on cartesian grids) to cases of non-uniform and non-aligned anisotropy fields $b$ and the simple extension to the case of a non-constant anisotropy intensity $1/\eps$. The mathematical approach and the numerical scheme are different from those presented in the previous work [Degond et al. (2010), arXiv:1008.3405v1] and its considerable advantages are pointed out.

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.

Francis Filbet, Claudia Negulescu, Chang Yang

This paper is devoted to the numerical approximation of a nonlinear temperature balance equation, which describes the heat evolution of a magnetically confined plasma in the edge region of a tokamak. The nonlinearity implies some numerical difficulties, in particular long time behavior, when solved with standard methods. An efficient numerical scheme is presented in this paper, based on a combination of a directional splitting scheme and the IMEX scheme introduced in [Filbet and Jin]

Anton Arnold, Claudia Negulescu

This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid "turning points". We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests.

Alexei Lozinski, Jacek Narski, Claudia Negulescu

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge-Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter $0 < \eps <1$, and this for fixed coarse Cartesian grids and for variable anisotropy directions. The context of this work are magnetically confined fusion plasmas.

Raffaele Carlone, Rodolfo Figari, Claudia Negulescu

We examine the suppression of quantum beating in a one dimensional non- linear double well potential, made up of two focusing nonlinear point interactions. The investigation of the Schrödinger dynamics is reduced to the study of a system of coupled nonlinear Volterra integral equations. For various values of the geometric and dynamical parameters of the model we give analytical and numerical results on the way states, which are initially confined in one well, evolve. We show that already for a nonlinearity exponent well below the critical value there is complete suppression of the typical beating behavior characterizing the linear quantum case.

Andrea Mentrelli, Claudia Negulescu

The electric potential is an essential quantity for the confinement process of tokamak plasmas, with important impact on the performances of fusion reactors. Understanding its evolution in the peripheral region - the part of the plasma interacting with the wall of the device - is of crucial importance, since it governs the boundary conditions for the burning core plasma. The aim of the present paper is to study numerically the evolution of the electric potential in this peripheral plasma region. In particular, we are interested in introducing an efficient Asymptotic-Preserving numerical scheme capable to cope with the strong anisotropy of the problem as well as the non-linear boundary conditions, and this with no huge computational costs. This work constitutes the numerical follow-up of the more mathematical paper by C. Negulescu, A. Nouri, Ph. Ghendrih, Y. Sarazin, "Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics".

Baptiste Fedele, Claudia Negulescu, Stefan Possanner

We develop an asymptotic-preserving scheme to solve evolution problems containing stiff transport terms. This scheme is based to a micro-macro decomposition of the unknown, coupled with a stabilization procedure. The numerical method is applied to the Vlasov equation in the gyrokinetic regime and to the Vlasov-Poisson 1D1V equation, which occur in plasma physics. The asymptotic-preserving properties of our procedure permit to study the long-time behavior of these models. In particular, we limit drastically by this method the numerical pollution, appearing in such time asymptotics when using classical numerical schemes.

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.

Baptiste Fedele, Claudia Negulescu

Goal of this paper is to investigate several numerical schemes for the resolution of two anisotropic Vlasov equations. These two toy-models arise from a kinetic description of a tokamak plasma confined by strong magnetic fields. The simplicity of our toy-models permits to better understand the features of each scheme, in particular to investigate their asymptotic-preserving properties, in the aim to choose then the most adequate numerical scheme for upcoming, more realistic simulations.

Alexei Lozinski, Jacek Narski, Claudia Negulescu

The main purpose of the present paper is to study from a numerical analysis point of view some robust methods designed to cope with stiff (highly anisotropic) elliptic problems. The so-called asymptotic-preserving schemes studied in this paper are very efficient in dealing with a wide range of $\varepsilon$-values, where $0 < \varepsilon \ll 1$ is the stiffness parameter, responsible for the high anisotropy of the problem. In particular, these schemes are even able to capture the macroscopic properties of the system, as $\varepsilon$ tends towards zero, while the discretization parameters remain fixed. The objective of this work shall be to prove some $\varepsilon$-independent convergence results for these numerical schemes and put hence some more rigor in the construction of such AP-methods.

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.