Convergence of splitting methods on rotating grids for the magnetized Vlasov equation

TL;DR

This work addresses efficient, accurate numerical solutions of the Vlasov equation with a constant background magnetic field by introducing a rotating velocity grid within a semi-Lagrangian framework and coupling to quasi-neutrality. A coordinate transform $\mathbf{v}=\mathbf{D}_{\omega_c}(t)\tilde{\mathbf{v}}$ removes the $\mathbf{v}\times\mathbf{B}_0$ rotation from velocity advection, enabling splitting in both the physical and rotating frames and a rigorous second-order convergence analysis via residuals $R(h)$ with $\|R(h)\|=O(h^3)$. Numerically, the rotating grid yields higher-fidelity solutions, better resolution of higher spatial/temporal modes (notably ion Bernstein waves), and reduced computational cost through merged steps and localized interpolation, demonstrated in nonlinear test cases and dispersion-growth studies. The approach offers a practical improvement for gyrokinetic-like electrostatic Vlasov simulations, enabling accurate large-time-step computations and robust unit testing within the BSL6D framework.

Abstract

Semi-Lagrangian solvers for the Vlasov system offer noiseless solutions compared to Lagrangian particle methods and can handle larger time steps compared to Eulerian methods. In order to reduce the computational complexity of the interpolation steps, it is common to use a directional splitting. However, this typically yields the wrong angular velocity. In this paper, we analyze a semi-Lagrangian method that treats the $v \times B$ term with a rotational grid and combines this with a directional splitting for the remaining terms. We analyze the convergence properties of the scheme both analytically and numerically. The favorable numerical properties of the rotating grid solution are demonstrated for the case of ion Bernstein waves.

Figures (5)

• Figure 1: Visualization of the solution of \ref{['num:equ:VlasovEquationRotationPart']} with a classical Strang-Splitting and on a rotating grid in the physical domain at $t=5/\omega_c$. The figures a) and b) give the full solution while c) and d) show the difference of the simulation results and the analytical solution in \ref{['num:equ:VlasovEquationRotationPartAnalyticalSolutionStrang', 'num:equ:VlasovEquationRotationPartAnalyticalSolutionRotatingGrid']}
• Figure 2: Difference of the analytical solution to simulation results for the test cases in \ref{['subsec:NumericalExampleVlasovEquationVxB', 'subsec:NumericalExampleVlasovEquationConstBgFields']} based on the L2 norm using $h=0.01/\omega_c$. The error is normalized with respect to the perturbation of the analytical solution $\delta f_\text{analytical}$. As is shown in \ref{['num:fig:rotatingGridComparison']} the rotating grid has no numerical error for the $\textbf{v} \times \textbf{B}_0$ simulation, which is therefore omitted in the plot.
• Figure 3: Convergence rates for integrators of \ref{['subsec:IntegratorsForTheVlasovEquation', 'subsec:IntegratorsForTheVlasovEquationRotatingFrame', 'subsec:NumericalExampleVlasovEquationConstBgFields']}. The convergence rate in the legend always omit the first data point of the measurement. The errors are estimated against a solution that has been obtained using a significantly smaller time step $h=0.0025/\omega_c$ and is referred to as a converged solution. The comparison carried out at $t=9.0/\omega_c$. The difference is normalized on the perturbation $\delta f$ of the converged solution.
• Figure 4: Dispersion relation of neutralizing ion Bernstein waves with a) fixed velocity grid and b) rotating velocity grid. The black dashed lines represent the analytical solution of the example. The length scale is normalized to the Larmor radius $\rho_L$.
• Figure 5: Growth rate of neutralizing ion Bernstein waves with a) fixed velocity grid and Strang splitting and b) rotating velocity grid. The solid lines represent the analytical solution of the growth rates for the first six harmonics of the IBWs with increasing frequency from light to dark green.