A split-step Active Flux method for the Vlasov-Poisson system

TL;DR

The paper tackles accurate, structure-preserving simulation of the Vlasov–Poisson system for collisionless plasmas using split-step Active Flux (AF) methods. It develops and compares multiple high-order flux evaluation strategies, including second- and third-order flux integrals and a discrepancy-distribution approach, all within directional operator-splitting frameworks (Lie, Strang, Yoshida). A consistent Poisson solver is integrated on AF grids to obtain the electric field, and extensive 1D1V tests (Landau damping and Two-Stream) demonstrate convergence, long-time fidelity, and conservation properties, with AF showing reduced dissipation compared to a state-of-the-art PFC method. The results indicate that split-step AF can deliver high-accuracy, scalable Vlasov simulations suitable for massively parallel architectures, offering a path toward 6D Vlasov–Maxwell solvers and integration into the MuPhy II ecosystem.

Abstract

Active Flux is a modified Finite Volume method that evolves additional Degrees of Freedom for each cell that are located on the interface by a non-conservative method to compute high-order approximations to the numerical fluxes through the respective interface to evolve the cell-average in a conservative way. In this paper, we apply the method to the Vlasov-Poisson system describing the time evolution of the time-dependent distribution function of a collisionless plasma. In particular, we consider the evaluation of the flux integrals in higher dimensions. We propose a dimensional splitting and three types of formulations of the flux integral: a one-dimensional reconstruction of second order, a third-order reconstruction based on information along each dimension, and a third-order reconstruction based on a discrepancy formulation of the Active Flux method. Numerical results in 1D1V phase-space compare the properties of the various methods.

Figures (12)

• Figure 1: Two-dimensional Active Flux grid arrangements: a) unsplit (left), b) split discrepancy distribution (middle), c) split (right)
• Figure 2: Naive Splitting for updating every DOF of a two-dimensional Active Flux cell in phase space. This approach results in a second-order accurate approximation of the flux-integral. The evolution operators $L_X$ (left) and $L_V$ (right) are applied on one-dimensional slices of the grid along each direction. Consequently, the resulting one-dimensional Active Flux cells within these slices consist of point values on the nodes and one-dimensional line averages on the edges or line averages and two-dimensional cell averages respectively. The cell averages of any cell are conservatively updated along both directions.
• Figure 3: Schematic update procedure for the two-dimensional cell-average. The values located on the $x$-edge at $v_{j-1/2}$ are updated by one-dimensional tracing along the $v$-direction to the time levels $t^{n+1/2}$ and $t^{n+1}$. The interface flux $H_{i, j-1/2}$ for that same edge is obtained by integration over the $x$-$t$-plane using the information of the electric field at time $t^n$.
• Figure 4: Exemplary one-dimensional updates in the $v$-dimension for computing the third-order fluxintegral over an edge of a two-dimensional AF cell. Characteristic tracing is indicated by the dotted lines: (left) on the edges along the $v$-direction, with conservation update of the line average of that edge, (right) on the centers of the $x$-edges without conservation update of the cell average which is carried out by computing the flux integral over the entire $x$-edge in space and time.
• Figure 5: Convergence of error for the initial phase of the weak Landau damping problem. Integrals were carried out analytically (left) or numerically by Simpsons rule (right) for schemes that require initial averaging of the distribution function.