# Multidimensional VlasovPoisson Simulations with High-order Monotonicity- and Positivity-preserving Schemes

## TL;DR

This paper introduces high-order, positivity-preserving numerical schemes for Vlasov-Poisson equations, improving accuracy and efficiency in simulating collisionless plasma and gravitational systems in high-dimensional phase space.

## Contribution

The authors develop and implement high-order monotonicity- and positivity-preserving schemes with semi-Lagrangian integration for Vlasov-Poisson equations, achieving better accuracy without increased computational cost.

## Key findings

- High-order schemes significantly improve accuracy over previous third-order methods.
- Semi-Lagrangian integration maintains computational efficiency.
- Successful large-scale 6D simulations on parallel computers.

## Abstract

We develop new numerical schemes for Vlasov--Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that positivity of the distribution function is also preserved. We adopt an efficient semi-Lagrangian time integration scheme that is more accurate and computationally less expensive than the three-stage TVD Runge-Kutta integration. We apply our spatially fifth- and seventh-order schemes to a suite of simulations of collisionless self-gravitating systems and electrostatic plasma simulations, including linear and nonlinear Landau damping in one dimension and Vlasov--Poisson simulations in a six-dimensional phase space. The high-order schemes achieve a significantly improved accuracy in comparison with the third-order positive-flux-conserved scheme adopted in our previous study. With the semi-Lagrangian time integration, the computational cost of our high-order schemes does not significantly increase, but remains roughly the same as that of the third-order scheme. Vlasov--Poisson simulations on $128^3 \times 128^3$ mesh grids have been successfully performed on a massively parallel computer.