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Numerical study of an anisotropic Vlasov equation arising in plasma physics

  • * Corresponding author: C. Negulescu

    * Corresponding author: C. Negulescu
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  • 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.

    Mathematics Subject Classification: Primary: 35Q83, 65M06; Secondary: 65F05.

    Citation:

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  • Figure 1.  Representation of the initial condition $f_{in}$ (A) and the exact solution $f_{ex}^{\epsilon}$ at the final time $T = 1$ (B). Here $\epsilon = 1$.

    Figure 2.  Representation of the exact limit solution $f^0_{ex}(t,x)$ at the final time $T$.

    Figure 3.  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).

    Figure 4.  Representation of the numerical solution $f^{\epsilon}$ for two values of $\epsilon$, and at the final time $T$, corresponding to the IMEX scheme.

    Figure 5.  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 $\epsilon = 1$.

    Figure 6.  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 $\epsilon = 1$.

    Figure 7.  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 $N_x = N_y = 1 001$).

    Figure 8.  Evolution of $\eta_\epsilon(T)$ and $\gamma_\epsilon(T)$ as a function of $\epsilon$ for each scheme.

    Figure 9.  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.

    Figure 10.  Condition number $cond(A)$ as a function of $\epsilon$ in log-log scale. The two curves correspond to the IMP and Lagrange-multiplier schemes.

    Figure 12.  Representation of the function $f^{\epsilon}$ at the final time $T$ for the IMP and Lagrange-multiplier scheme, with several values of $\epsilon$.

    Figure 11.  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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