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.
Highlights
Numerical simulations are of key importance for the understanding of the behavior of plasmas in a nuclear fusion device
We distinguish two classes of grid-based methods, Eulerian solvers based on finite volume or discontinuous Galerkin on the one hand, and, on the other hand, semi-Lagrangian methods that update the solution by evolution along the characteristics using interpolation
We focus on efficient parallelization schemes for a semi-Lagrangian discretization of the Vlasov–Poisson equation in 6-D phase space
Summary
Numerical simulations are of key importance for the understanding of the behavior of plasmas in a nuclear fusion device. The idea of the remapping scheme is to work with two different domain partitions which both keep a partition of the dimensions sequential on each processor The latter strategy is very well adapted to a semi-Lagrangian method combined with dimensional splitting; its parallel scalability is hampered by an all-to-all communication pattern. We demonstrate that the restriction can be alleviated by an asymmetric communication scheme This problem is severe in simulations of magnetized plasmas in a strong guide field where particles perform a fast gyromotion around the magnetic field lines.
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