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.
Highlights
The kinetic description of a plasma is based on the Vlasov equation, which is a partial differential equation satisfied by a distribution function of the charged particles depending on the time t ≥ 0, the space variable x ∈ Rd and the velocity v ∈ Rd
The so-obtained numerical scheme satisfies the following interesting properties: (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 consistent approximation of the limiting equation in the fluid regime
We construct a scheme with the following two properties: (i) the usual noise which is observed in standard PIC method is strongly reduced by the micro-macro decomposition, (ii) the PIC method is used only for the micro part and the number of particles can be very small in the fluid regime
Summary
The kinetic description of a plasma is based on the Vlasov equation, which is a partial differential equation satisfied by a distribution function of the charged particles depending on the time t ≥ 0, the space variable x ∈ Rd and the velocity v ∈ Rd. An important input in this work is the fact that the kinetic part of the micro-macro model is discretized by using particle method, whereas a grid of the phase space is used in [3] In this way, we construct a scheme with the following two properties: (i) the usual noise which is observed in standard PIC method is strongly reduced by the micro-macro decomposition, (ii) the PIC method is used only for the micro part and the number of particles can be very small in the fluid regime. We describe our strategy to solve the micro-macro system (1)-(2)-(3) As said before, this is done by following three steps: numerical resolution of (11) using particles, matching the moments of g to zero and numerical resolution of (12) using a finite volume method.
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