Splitting scheme for gyro-kinetic equations with Semi-Lagrangian and Arakawa substeps

TL;DR

A new algorithm for solving the gyro-kinetic model combining the Semi-Lagrangian method and the Arakawa (AKW) scheme with a time-integrator is proposed, to improve the conservation of the physical constants.

Abstract

The gyro-kinetic model is an approximation of the Vlasov-Maxwell system in a strongly magnetized magnetic field. We propose a new algorithm for solving it combining the Semi-Lagrangian (SL) method and the Arakawa (AKW) scheme with a time-integrator. Both methods are successfully used in practice for different kinds of applications, in our case, we combine them by first decomposing the problem into a fast (parallel) and a slow (perpendicular) dynamical system. The SL approach and the AKW scheme will be used to solve respectively the fast and the slow subsystems. Compared to the scheme in [1], where the entire model is solved using only the SL method, our goal is to replace the method used in the slow subsystem by the AKW scheme, in order to improve the conservation of the physical constants.

Figures (8)

• Figure 1: The nine (left) and thirteen (right) point stencil, where the $+$-parts are highlighted in solid orange, while the $\times$-parts are in dashed blue. Here, the middle of the cell corresponds to the point on which a stencil is defined.
• Figure 2: The density $f$ at initial time.
• Figure 3: Convergence curve of the $L^2$-error from Table \ref{['tab:num_exp']}.
• Figure 4: The relative error between two time-steps of the conserved quantities (left), i.e. equation \ref{['eq:pol_cons_quant']}, and their algebraic indicators at that time-step (right), i.e. equation \ref{['eq:alg_prop']}.
• Figure 5: The $L^2$-norm of the electric potential $\phi$ comparing the Arakawa method and the Semi-Lagrangian scheme. The results of both simulations closely follow the analytical growth rate of $\norm{\phi}_2 = 4\cdot 10^{-5} \times \exp\left(0.00354 t\right)$ in the linear regime (until $t\sim 3000$) and then saturate in the order of magnitude 10. Observe that the left plot is with a semi-logarithmic scale on the $y$-axis, while the right plot is linear on both axes.