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

TL;DR

This paper analyzes how splitting methods for the Vlasov-HMF model exhibit nonlinear Landau damping, showing that numerical solutions scatter to a modified state with polynomial damping rate, closely approximating the continuous solution.

Contribution

It provides the first rigorous proof of nonlinear Landau damping for splitting methods applied to the Vlasov-HMF model, including error estimates and stability analysis.

Findings

Numerical solutions scatter to a modified state with polynomial damping.

The modified state remains close to the continuous solution.

Error estimates relate the damping rate to the time step size.

Abstract

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.