We develop an asymptotic-preserving scheme to solve evolution problems containing stiff transport terms. This scheme is based to a micro-macro decomposition of the unknown, coupled with a stabilization procedure. The numerical method is applied to the Vlasov equation in the gyrokinetic regime and to the Vlasov-Poisson 1D1V equation, which occur in plasma physics. The asymptotic-preserving properties of our procedure permit to study the long-time behavior of these models. In particular, we limit drastically by this method the numerical pollution, appearing in such time asymptotics when using classical numerical schemes.

Goal of this paper is to investigate several numerical schemes for the resolution of two anisotropic Vlasov equations. These two toy-models are obtained from a kinetic description of a tokamak plasma confined by strong magnetic fields. The simplicity of our toy-models permits to better understand the features of each scheme, in particular to investigate their asymptotic-preserving properties, in the aim to choose then the most adequate numerical scheme for upcoming, more realistic simulations.

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