Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

Anaïs Crestetto; Nicolas Crouseilles; Mohammed Lemou

doi:10.3934/krm.2012.5.787

Article Contents

2012, Volume 5, Issue 4: 787-816. Doi: 10.3934/krm.2012.5.787

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Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

1.
INRIA-Nancy Grand Est, CALVI Project, and IRMA, Université de Strasbourg, 67084, STRASBOURG
2.
INRIA-Rennes Bretagne Atlantique, IPSO Project, and IRMAR (Université de Rennes 1), 35042 RENNES
3.
CNRS and IRMAR (Université de Rennes 1), and INRIA-Rennes Bretagne Atlantique, IPSO Project, 35042 RENNES

Received: March 31, 2012

Revised: May 31, 2012

Published: November 2012

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Abstract

This work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.

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Mathematics Subject Classification: 65M06, 35B25, 82C80, 82D10, 41A60.

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