Jacek Narski

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.

Jacek Narski, Maurizio Ottaviani

This paper deals with the numerical study of a strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. Furthermore, the recently proposed Asymptotic-Preserving method [arXiv:1203.6739] allows to perform simulations regardless of the anisotropy strength but its application is limited to the case, where the anisotropy direction is given by a field with all field lines open. In this paper we introduce a new Asymptotic-Preserving method, which overcomes those limitations without any loss of precision or increase in the computational costs. 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.

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.

Jacek Narski

The concern of this work is the generalization of an Asymptotic Preserving method for the highly anisotropic elliptic equations presented in [P. Degond, A. Lozinski, J. Narski, and C. Negulescu. An asymptotic-preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition. J. Comput. Phys., 231(7):2724{2740, 2012]. The limitations of the method introduced there in are omitted by the introduction of a stabilization term in the Asymptotic Reformulation. Furthermore, anisotropic error indicators and mesh adaptation algorithms are proposed and tested allowing to reduce considerably the number of mesh points required to achieve prescribed precision. Reported meshes have maximum aspect ratio greater than 500.

Jacek Narski

In this paper we present a parallelization strategy on distributed memory systems for the Fast Kinetic Scheme --- a semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] for solving kinetic equations. The original algorithm was proposed for the BGK approximation of the collision kernel. In this work we deal with its extension to the full Boltzmann equation in six dimensions, where the collision operator is resolved by means of fast spectral method. We present close to ideal scalability of the proposed algorithm on tera- and peta-scale systems.

Giacomo Dimarco, Raphaël Loubère, Jacek Narski et al.

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the $3$D$\times 3$D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

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.