# A "metric" semi-Lagrangian Vlasov-Poisson solver

## TL;DR

This paper introduces a novel semi-Lagrangian Vlasov-Poisson solver that uses metric elements to accurately track phase-space flow, enabling precise reconstruction of the distribution function in gravitational and plasma dynamics.

## Contribution

The paper presents a new metric-based semi-Lagrangian approach that improves phase-space tracking and reconstruction in Vlasov-Poisson simulations, with potential for high-dimensional applications.

## Key findings

- Accurate phase-space reconstruction using second-order geometry expansion.
- Efficient remapping with higher-order splines during transport.
- Algorithm successfully tested in 1D gravitational dynamics, extendable to higher dimensions.

## Abstract

We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements of metric to follow locally the flow and its deformation, allowing one to find quickly and accurately the initial phase-space position $Q(P)$ of any test particle $P$, by expanding at second order the geometry of the motion in the vicinity of the closest element. It is thus possible to reconstruct accurately the phase-space distribution function at any time $t$ and position $P$ by proper interpolation of initial conditions, following Liouville theorem. When distorsion of the elements of metric becomes too large, it is necessary to create new initial conditions along with isotropic elements and repeat the procedure again until next resampling. To speed up the process, interpolation of the phase-space distribution is performed at second order during the transport phase, while third order splines are used at the moments of remapping. We also show how to compute accurately the region of influence of each element of metric with the proper percolation scheme. The algorithm is tested here in the framework of one-dimensional gravitational dynamics but is implemented in such a way that it can be extended easily to four or six-dimensional phase-space. It can also be trivially generalised to plasmas.