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

## TL;DR

This paper develops a scalable semi-Lagrangian solver for the six-dimensional Vlasov-Poisson equation, addressing high memory demands and parallelization challenges through innovative schemes, enabling efficient computation on supercomputers.

## Contribution

It introduces a parallelization approach with a blocked communication scheme and pipelining for the 6d Vlasov-Poisson solver, improving scalability and efficiency.

## Key findings

- Achieved parallel scalability on up to 65,000 processes.
- Found the domain decomposition approach superior for massively parallel cases.
- Demonstrated effective handling of artificial time step restrictions.

## Abstract

This paper 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 paper, we consider the 6d Vlasov-Poisson problem discretized by a split-step semi-Lagrangian scheme, using successive 1d interpolations on 1d stripes of the 6d domain. Two parallelization paradigms are compared, a remapping scheme and a classical domain decomposition approach applied to the full 6d 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 65k processes is demonstrated for benchmark problems on a supercomputer.