Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

Pages: 887 – 911

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n4.a1

Authors

Anaïs Crestetto (Laboratoire de Mathématiques Jean Leray, Université de Nantes, France; and INRIA Rennes-Bretagne Atlantique, Rennes, France)

Nicolas Crouseilles (INRIA Rennes-Bretagne Atlantique, Rennes, France; and Institut de Recherche Mathématiques, Université de Rennes, France)

Mohammed Lemou (Institut de Recherche Mathématiques, Université de Rennes, France; and INRIA Rennes-Bretagne Atlantique, Rennes, France)

Abstract

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevents from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step $\Delta t$ in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [Crestetto, Crouseilles, Lemou, Kin. Rel. Models, 5:787–816, 2012] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit.

Keywords

kinetic models, asymptotic preserving scheme, diffusive scaling, particle-in-cell, micro-macro decomposition

2010 Mathematics Subject Classification

35B25, 35Q83, 65M75, 82D10

Full Text (PDF format)

N. Crouseilles and M. Lemou are supported by the French ANR project MOONRISE ANR-14-CE23-0007-01 and by the Enabling Research EUROFusion project CfP-WP14-ER-01/IPP-03. A. Crestetto is supported by the French ANR project ACHYLLES ANR-14-CE25-0001 and by the French ANR project MoHyCon ANR-17-CE40-0027-01.

Received 7 January 2017

Accepted 15 January 2018

Published 31 October 2018