# American Institute of Mathematical Sciences

December  2018, 11(6): 1395-1426. doi: 10.3934/krm.2018055

## Numerical study of an anisotropic Vlasov equation arising in plasma physics

 Université de Toulouse & CNRS, UPS, Institut de Mathématiques de Toulouse UMR 5219, F-31062, Toulouse, France

* Corresponding author: C. Negulescu

Received  October 2016 Revised  July 2017 Published  June 2018

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.

Citation: Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic & Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055
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##### References:
Representation of the initial condition $f_{in}$ (A) and the exact solution $f_{ex}^{\epsilon}$ at the final time $T = 1$ (B). Here $\epsilon = 1$.
Representation of the exact limit solution $f^0_{ex}(t,x)$ at the final time $T$.
Time-evolution of the exact solution at point $(x_{N_x-1},y_{N_y-1})$ in the two dimensional case (A) with $T = 12$ and $N_t = 501$; resp. at point $y_{N_y-1}$ in the one dimensional case with $T = 10$, $a = 0$ and $N_t = 501$ (B).
Representation of the numerical solution $f^{\epsilon}$ for two values of $\epsilon$, and at the final time $T$, corresponding to the IMEX scheme.
Left (A): Plot of the num. sol. $f^{\epsilon}$ for $\epsilon = 10^{-10}$, at the final time $T$. Right (B): Time-evolution of the IMEX scheme sol. at point $y_{N_y-1}$ in the 1D case for $T = 10$ and several $\epsilon$. We have added the exact solution for $\epsilon = 1$.
Time-evolution of the solution via Fourier (A) and IMEX, MM- resp. Lagrange-multiplier schemes (B), at $y_{N_y-1}$ in 1D with $T = 10$, $a = 0$, $N_t = 501$. We have added in both cases the exact solution for $\epsilon = 1$.
Evolution of the $L^{\infty}$-error between $f^{\epsilon}_{ex}(t,\cdot)$ and $f^{\epsilon}(t,\cdot)$ at final time $T = 1$ and for $\epsilon = 1$, as a function of $\Delta x$ (with $N_y = 15 001$, $N_t = 15 001$), $\Delta y$ (with $N_x = 15 001$, $N_t = 15 001$) and $\Delta t$ (with $N_x = N_y = 1 001$).
Evolution of $\eta_\epsilon(T)$ and $\gamma_\epsilon(T)$ as a function of $\epsilon$ for each scheme.
Condition number $cond(A)$ as a function of $\epsilon$ in log-log scale. The three curves correspond to the IMEX, Micro-Macro and Lagrange-multiplier schemes.
Condition number $cond(A)$ as a function of $\epsilon$ in log-log scale. The two curves correspond to the IMP and Lagrange-multiplier schemes.
Representation of the function $f^{\epsilon}$ at the final time $T$ for the IMP and Lagrange-multiplier scheme, with several values of $\epsilon$.
Representation of a cut at $x = 0$ of $f^{\epsilon}_{num}$ at the final time $T$ for the IMP and Lagrange-multiplier schemes, and several values of $\epsilon$.
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