A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov–Poisson equation

Abstract

This article presents an optimized and scalable semi-Lagrangian solver for the Vlasov–Poisson system in six-dimensional phase space. Grid-based solvers of the Vlasov equation are known to give accurate results. At the same time, these solvers are challenged by the curse of dimensionality resulting in very high memory requirements, and moreover, requiring highly efficient parallelization schemes. In this article, we consider the 6-D Vlasov–Poisson problem discretized by a split-step semi-Lagrangian scheme, using successive 1-D interpolations on 1-D stripes of the 6-D domain. Two parallelization paradigms are compared, a remapping scheme and a domain decomposition approach applied to the full 6-D problem. From numerical experiments, the latter approach is found to be superior in the massively parallel case in various respects. We address the challenge of artificial time step restrictions due to the decomposition of the domain by introducing a blocked one-sided communication scheme for the purely electrostatic case and a rotating mesh for the case with a constant magnetic field. In addition, we propose a pipelining scheme that enables to hide the costs for the halo communication between neighbor processes efficiently behind useful computation. Parallel scalability on up to 65,536 processes is demonstrated for benchmark problems on a supercomputer.