High-order Hamiltonian splitting for Vlasov-Poisson equations

Fernando Casas (1) , Nicolas Crouseilles (2, 3) , Erwan Faou (2, 3) , Michel Mehrenberger (4, 5, 6)

1 IMAC - Institut Universitari de Matemátiques i Aplicacions de Castelló
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IPSO - Invariant Preserving SOlvers
4 TONUS - TOkamaks and NUmerical Simulations
5 IPP - Max-Planck-Institut für Plasmaphysik [Garching]
6 IRMA - Institut de Recherche Mathématique Avancée

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.

Mots clés: Runge-Kutta-Nyström methods; Vlasov-Poisson equations; Hamiltonian splitting; high-order splitting

Domaines: Analyse numérique [math.NA]; Physique Numérique [physics.comp-ph]

Fernando Casas, Nicolas Crouseilles, Erwan Faou, Michel Mehrenberger. High-order Hamiltonian splitting for Vlasov-Poisson equations. Numerische Mathematik, 2017, 135 (3), pp.769-801. ⟨10.1007/s00211-016-0816-z⟩. ⟨hal-01206164⟩