Solving the six-dimensional Vlasov-Maxwell System with Active Flux and Splitting Methods

TL;DR

The paper develops a split-step Active Flux framework to solve the 6D Vlasov–Maxwell system efficiently by reducing dimensionality to 1D subproblems, leveraging AF's compact, high-order flux reconstruction. Coupled with a Yee-grid FDTD Maxwell solver and Boris correction, this approach achieves lower numerical dissipation and reduced anisotropy compared with a semi-Lagrangian PFC benchmark, while maintaining accurate kinetic dynamics. Across a suite of test problems—normalization, spherical blast waves, linear Landau damping, electron Bernstein waves, and the Orszag–Tang vortex—the AF method demonstrates superior resolution of phase-space structures and high-wavenumber modes at comparable or fewer degrees of freedom and typically lower compute cost. The work highlights practical considerations such as Gauss-law preservation and suggests future improvements in divergence cleaning and coupling strategies for large-scale kinetic-plasma simulations.

Abstract

Active Flux (AF) is a modified Finite Volume method that evolves additional Degrees of Freedom (DoF) located on the cell interfaces to compute high-order approximations to the numerical fluxes through the respective interface. We present an AF-based scheme for the simulation of collisionless plasmas described by the Vlasov equation coupled with Maxwell's equations. In order to limit the DoF in high dimensional settings we employ operator splitting. The resulting one-dimensional advection equations can be solved efficiently and with low implementation complexity, making it a very fast alternative to standard Finite Volume methods. We compare our scheme's performance with a related Finite Volume method based on the semi-Lagrangian approach. We find that, as a consequence of its compact stencil, the AF scheme has significantly lower dissipation and reduced anisotropy, and thus produces results on par with or even superior to the benchmark for standard test cases reproducing important kinetic phenomena, while also offering lower computational cost.

Figures (9)

• Figure 1: Left: Degrees of freedom in the 1D AF method. Center: Semi-Lagrangian update of the interface DoF by backtracing the characteristic. Right: Reconstruction of the solution to be evaluated at the footpoint of the characteristics.
• Figure 2: Splitting approach for a 2D AF cell in phase space. The evolution operators for the update in $x-$direction (left) and $v-$direction (right) are applied on (one-dimensional) fibers of the grid along each direction. For the 2D grid, this means the following: the corner DoF are point values and the center DoF are cell averages, but edge DoF are treated as line averages in direction of the respective edge. This is the result of a given edge value acting as the central average during the update in the direction parallel to its edge, but being treated as an interface point value during the update in normal direction. Figure as in Hensel2024.
• Figure 3: Cells of the split-step AF grid. Left: 2D. Corner values remain point values and center values remain cell averages, but edge values are line averages depending on their orientation. Right: 3D. The grid consists of point values on the cell corners (yellow), edge averages (green), face averages (blue), and the cell average (purple).
• Figure 4: A slice at the location of the respective DoF in configuration space of AF and PFC, respectively, as well as the chosen Yee-grid. For PFC, the Yee-cube coincides with the configuration space cell, while in the AF case, each DoF hosts its individual Yee-cube.
• Figure 5: Blast wave test at $t=0.05$. Left: AF with $64^2\times32^2$ cells, right: PFC with $64^2\times32^2$.