High-order Hamiltonian splitting for Vlasov-Poisson equations

TL;DR

This paper develops high-order Hamiltonian splitting methods for the Vlasov-Poisson equations, demonstrating their efficiency and providing rigorous convergence analysis and error estimates.

Contribution

It introduces new high-order splitting schemes based on Hamiltonian decomposition, with simplified order conditions and proven convergence for Vlasov-Poisson equations.

Findings

High-order schemes improve numerical accuracy.

Methods are efficient with fewer stages in 1D cases.

Numerical results confirm the benefits of high-order splitting.

Abstract

We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nystr{\"o}m type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.