High-order Hamiltonian splitting for the Vlasov–Poisson equations

Abstract

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-Nyström 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.

An exponential integrator for the drift-kinetic model

“…In this section, we discuss the numerical results of the second order exponential integrator described by (4). In Fig.…”

“…The reason for this is that (as has been realized in the seminal paper by Cheng & Knorr [5]) splitting schemes often decouple different parts of the equation and allow us to perform computations on lower-dimensional slices of phase space. A significant part of the literature has been devoted to the construction and analysis of splitting methods (see, for example, [4,5,12,13,22]). In addition, a range of space discretization schemes has been proposed.…”

A high order semi-Lagrangian discontinuous Galerkin method for Vlasov–Poisson simulations without operator splitting

“…One prominent advantage of performing operator splitting is that the resulting split equations are linear and in lower dimensions, thus largely simplifying the algorithm design and implementation of SL schemes [18,7]. Most existing high order SL schemes are designed based on the splitting strategy.…”

A comparison of boundary correction methods for Strang splitting

“…Consequently, a significant body of research has been devoted to splitting methods. Among their applications are diffusion-reaction equations [12,18,15,24,16], advection-reaction equations [23,28,24], diffusion-reaction-advection equations [11,24], Schrödinger-type equations [3,27,17], dispersive equations [25,22,14], and kinetic equations [7,19,13,9,6,10].…”