An asymptotic preserving scheme based on a micro-macrodecomposition for Collisional Vlasov equations: diffusion and high-field scaling limits

Nicolas Crouseilles; Mohammed Lemou

doi:10.3934/krm.2011.4.441

Article Contents

2011, Volume 4, Issue 2: 441-477. Doi: 10.3934/krm.2011.4.441

This issue Previous Article On a model for mass aggregation with maximal size Next Article On Villani's conjecture concerning entropy production for the Kac Master equation

An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits

Nicolas Crouseilles^1, and
Mohammed Lemou^2,

1.
INRIA-Nancy Grand Est and IRMA, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg
2.
CNRS and IRMAR, Université de Rennes 1, 263 Avenue du General Leclerc CS74205, 35042 Rennes cedex

Received: September 30, 2010

Revised: January 31, 2011

Published: May 2011

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Abstract

In this work, we extend the micro-macro decomposition based numerical schemes developed in [3] to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in [30], we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when $\varepsilon\rightarrow 0$, which makes it free from the usual diffusion constraint $\Delta t=O(\Delta x^2)$ in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.

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

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