Journal of Computational Physics 291 (2015) 99?119

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BG 1. flu fo us ou co fin pr so th by pr of ? of * vi ht 00Journal of Computational Physics www.elsevier.com/locate/jcp multiscale fast semi-Lagrangian method for rarefied gas ynamics? iacomo Dimarco a,b,?, Rapha?l Loub?re b, Vittorio Rispoli b epartment of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy niversit? de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut de Math?matiques de Toulouse; F-31062 Toulouse, France r t i c l e i n f o a b s t r a c t ticle history: ceived 7 March 2014 ceived in revised form 18 December 2014 cepted 9 February 2015 ailable online 6 March 2015 ywords: netic equations screte velocity models mi-Lagrangian schemes undary conditions main decomposition ultiscale problems

K equation

In this paper we consider the extension of the method developed in Dimarco and Loub?re (2013) [22,23] with the aim of facing the numerical resolution of multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere. This is the case of many realistic applications: some regions of the computational domain require a microscopic description, given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type. Our aim is to show how the kinetic scheme developed in the pre-cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi-scale realistic problems. Several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme. ? 2015 Elsevier Inc. All rights reserved.

Introduction

Kinetic models are intended to give a probabilistic description of particle systems. Concerning the case of gases and ids, this microscopic description is necessary whenever the system is far from its equilibrium state. This is often the case r rarefied gases and plasmas, for which the mean free path between two successive collisions is large enough to forbid the e of macroscopic models based on the hypothesis of local thermodynamical equilibrium [11]. Thus kinetic equations turn t to be very useful in describing many physical phenomena, however their use is limited by the excessive computational st when realistic simulations are considered. On the one side classical deterministic techniques such as finite volume, ite elements or spectral methods [37,39,40,27] are too costly when it is desirable to deal with the full seven dimensional oblem (six dimensional phase?space plus time). Recently it has been shown that the spectral approach [38] is able to lve, at least partially, this problem. On the other side, probabilistic Monte Carlo techniques [3,10] are often used to tame e curse of dimensionality. However, these methods are affected by many fluctuations which are commonly attenuated time averages for steady state situations while, for unsteady problems, the reduction of the variance is still an open oblem [15,21] with apparently no easy solution. In order to reduce the statistical noise, in practice an enormous number samples is generally used and this leads again to very expensive simulations.

This work was supported by the ANR Blanc project ?BOOST?, by the ANR JCJC project ?ALE INC(ubator) 3D? and by the ?Bando Giovani Ricercatori 2013?

Ferrara University (Italy).

Corresponding author.

E-mail addresses: giacomo.dimarco@math.univ-toulouse.fr, giacomo.dimarco@unife.it (G. Dimarco), raphael.loubere@math.univ-toulouse.fr (R. Loub?re), torio.rispoli@math.univ-toulouse.fr (V. Rispoli). tp://dx.doi.org/10.1016/j.jcp.2015.02.031 21-9991/? 2015 Elsevier Inc. All rights reserved. 100 G. Dimarco et al. / Journal of Computational Physics 291 (2015) 99?119

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Al diIn a recent paper [22] the authors have developed a new deterministic method to solve kinetic equations named Fast netic Scheme (FKS). In particular they have designed an efficient technique for the discretization of the linear transport rt of these equations. The proposed method is based on the so-called Discrete Velocity Model (DVM) [37] and on a mi-Lagrangian approach [13,14]. The DVM is obtained by a direct discretization of the velocity space into a finite set of ed velocities [5,37,39]. As a result of such discretization the original kinetic equation is then represented as a system of ear transport equations plus an interaction term coupling all equations and corresponding to the discretization of the llision operator. The purpose of the method described in [22,23] was to drastically reduce the cost of the transport part the discrete velocity model using a semi-Lagrangian technique, allowing to exactly solve the transport part on the entire main at a negligible cost. Hence, the cost of the solution of the original kinetic equation was almost entirely due to the ojection of the solution onto the grid to compute the collision operator. The resulting scheme shares many analogies with assical semi-Lagrangian methods [28,13,14] and Monte Carlo schemes [3]. However, contrarily to the latter, the scheme as fast as a particle method while providing a fully deterministic numerical solution, not affected by any source of atistical error. The authors have shown that the full six dimensional problem (space and velocity) can be solved on a ngle processor laptop on decent meshes for a simple kinetic equation consisting of transport plus a relaxation operator. owever such scheme still remains very expensive for a single processor machine if space or velocity grids need to be larged to treat more realistic problems. Consequently parallel versions of this scheme have been designed in [24].