Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths, and is close to a convection CFL condition for large mean free paths. Ou r analysis is based on very simple energy estimates

A Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects

This paper collects the efforts done in our previous works [P. Degond, S. Jin, L. Mieussens, A Smooth Transition Between Kinetic and Hydrodynamic Equations, J. Comp. Phys., 209 (2005) 665--694.],[P.Degond, G. Dimarco, L. Mieussens, A Moving Interface Method for Dynamic Kinetic-fluid Coupling, J. Comp. Phys., Vol. 227, pp. 1176-1208, (2007).],[P. Degond, J.G. Liu, L. Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi. Model. Sim. 5(3), 940--979 (2006)] to build a robust multiscale kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move, merge, appear or disappear in time. The main ingredients of the present work are the followings ones: a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary; this multiscale description of the flow is obtained by using a micro-macro decomposition of the distribution function [P. Degond, J.G. Liu, L. Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi. Model. Sim. 5(3), 940--979 (2006)]; the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function; to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself. Several numerical examples are presented to validate the method and measure its computational efficiency.Comment: 34 page

A hybrid method for hydrodynamic-kinetic flow - Part I -A particle-gridmethod for reducing stochastic noise in kinetic regimes

In this work we present a hybrid particle-grid Monte Carlo method for the Boltzmann equation, which is characterized by a significant reduction of the stochastic noise in the kinetic regime. The hybrid method is based on a first order splitting in time to separate the transport from the relaxation step. The transport step is solved by a deterministic scheme, while a hybrid DSMC-based method is used to solve the collision step. Such a hybrid scheme is based on splitting the solution in a collisional and a non-collisional part at the beginning of the collision step, and the DSMC method is used to solve the relaxation step for the collisional part of the solution only. This is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution, performing collisions as in a standard DSMC method, and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution. The latter is added to a piecewise constant reconstruction of the non-collisional part of the solution, which in fact remains unchanged during the relaxation step. Numerical results show that the stochastic noise is significantly reduced at large Knudsen numbers with respect to the standard DSMC method. Indeed in this algorithm, the particle scheme is applied only on the collisional part of the solution, so only this fraction of the solution is affected by stochastic fluctuations. But since the collisional part of the solution reduces as the Knudsen number increases, stochastic noise reduces as well at large Knudsen number

Schémas explicites à pas de temps local pour la CFD dans un contexte Volumes Finis

International audienceIn this paper, two local time stepping schemes of order two and three in time are proposed. By construction, they are not mass conservative but a correction stage is added to make them conservative. These schemes are compared with some local time stepping schemes existing in the literature (schemes of Constantinescu and Sandu). The comparisons are carried out on various test cases. They prove that our schemes are efficient and our third order local time stepping has a higher time accuracy than the schemes based on the strategy of Constantinescu and Sandu. Our third order local time stepping scheme is used to perform an industrial-like test case: a 3D Large-Eddy Simulation over an airfoil.Dans ce papier, deux schémas à pas de temps local d'ordre deux et trois en temps sont proposés. Par construction, ils ne sont pas conservatifs mais une étape de correction est ajoutée pour les rendre conservatifs. Ces schémas sont comparés à des schémas à pas de temps local de la littérature (schémas de Constantinescu et Sandu). Les comparaisons sont réalisées sur des cas-test variés, et montrent que nos schémas sont efficaces. Notre schéma à pas de temps local d'ordre trois se révèle plus précis que les schémas de Constantinescu et Sandu. Il est utilisé pour réaliser un cas-test de type industriel : une simulation LES, 3D, d'un écoulement autour d'un profil d'aile

Un réseau qui doit s’organiser : les 170 musées de l’école en France

Au premier bouleversement de l’opinion, entre 1880 et 1900, caractérisé par l’enseignement primaire, gratuit et obligatoire, avec ses bâtiments nouveaux y compris dans les hameaux et son nouveau matériel pédagogique, correspond un second bouleversement de l’opinion dans les années 1960-1975 avec une scolarité obligatoire jusqu’à 16 ans, des bâtiments et des matériels qui se veulent modernes et une pédagogie renouvelée.Les conséquences de cette évolution brutale et décisive sont la nostalgie d’une génération d’anciens élèves et la volonté de garder et de faire vivre un patrimoine payé par les communes et utilisé par les citoyens. Nous assistons à la floraison de cent soixante-dix musées de l’école ou de musées d’Arts et Traditions populaires, dans lesquels l’école prend toute sa place. C’est un mouvement d’opinion d’une grande ampleur qui touche la presque totalité des départements.Il faut lire la thèse de Jean-François Chanet soutenue en 1994 « L’école républicaine et les petites patries » et le monumental ouvrage publié en 1999 « Le Patrimoine de l’Éducation Nationale », Pierre Caspard, directeur scientifique. Ils permettent de comprendre la diversité de la France, l’extraordinaire diversité des musées de l’école.Il appartient à l’Association des Amis du Musée national de l’Éducation de rompre leur isolement, de les aider

Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation

In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the author's knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D

Microscopically implicit-macroscopically explicit schemes for the BGK equation

In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condi- tion on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of ener- getic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched

An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

International audienceIn this paper we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. This scheme is uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods

An efficient numerical method for solving the Boltzmann equation in multidimensions

International audienceIn 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 3D×3D 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