On numerical Landau damping for splitting methods applied to the Vlasov-HMF model

Erwan Faou (1) , Romain Horsin (1) , Frédéric Rousset (2)

1 MINGUS - Multi-scale numerical geometric schemes
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider time discretizations of the Vlasov-HMF (Hamiltonian Mean-Field) equation based on splitting methods between the linear and non-linear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time stepsize.

Mots clés: HMF model; Geometric Numerical Integration; Splitting methods; Vlasov equations; Landau damping

Domaines: Analyse numérique [math.NA]

Erwan Faou, Romain Horsin, Frédéric Rousset. On numerical Landau damping for splitting methods applied to the Vlasov-HMF model. Foundations of Computational Mathematics, 2018, 18 (1), pp.97-134. ⟨10.1007/s10208-016-9333-9⟩. ⟨hal-01219115⟩