Publikationen
Lukas Hensel, Yanick Kiechle, Rainer Grauer, Gudrun Grünwald, Christiane Helzel, Katharina KormannA Comparison of Active Flux Methods for the Vlasov-Poisson SystemActive Flux is a third-order accurate, fairly novel finite volume method for hyperbolic conservation laws that is becoming increasingly popular. It evolves additional nodal degrees of freedom (DOF) located on cell interfaces and shared by neighboring cells. The numerical fluxes are then computed from these DOFs. A crucial component of Active Flux methods is the evolution operator of the point values, which enables the natural use of semi-Lagrangian ideas and makes Active Flux an attractive candidate for a grid-based approach to the Vlasov equation. Here, we compare two recently proposed Active Flux methods for the 1D1V Vlasov-Poisson system: a split-step method and an unsplit method. accepted and to appear in ENUMATH 2025 Proceedings (2026) 
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Lukas Hensel, Gudrun Grünwald, Katharina Kormann, Rainer GrauerA split-step Active Flux method for the Vlasov-Poisson systemActive 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. J. Comp. Phys. 540 (2025) 114294  
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G. Grünwald, L. Hensel, M. Deisenhofer, S. Lautenbach, K. Kormann, R. GrauerSolving the six-dimensional Vlasov–Maxwell System with Active Flux and Splitting MethodsActive 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. submitted (2025)
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