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

TL;DR

This paper analyzes a semi-Lagrangian method with a rotating grid for the magnetized Vlasov equation, demonstrating its convergence and effectiveness in simulating ion Bernstein waves, thus improving computational efficiency and accuracy.

Contribution

It introduces and analyzes a novel semi-Lagrangian scheme using a rotating grid to handle the magnetic term in the Vlasov equation, addressing issues with traditional directional splitting.

Findings

The scheme converges both analytically and numerically.

It accurately simulates ion Bernstein waves.

The rotating grid improves computational efficiency.

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.