Numerical resolution of the Vlasov equation for the Hamiltonian Mean-Field model

TL;DR

This paper conducts detailed numerical simulations of the Vlasov equation for the Hamiltonian Mean-Field model, analyzing stability, numerical methods, and limitations of the approach for systems with long-range interactions.

Contribution

It provides a comprehensive numerical analysis of the Vlasov equation for the Hamiltonian Mean-Field model, including stability checks and evaluation of the semi-Lagrangian algorithm.

Findings

Validation of stability results for the homogeneous state

Assessment of the semi-Lagrangian algorithm's numerical properties

Identification of limitations due to finite resolution

Abstract

We present in this paper detailed numerical Vlasov simulations of the Hamiltonian Mean-Field model. This model is used as a representative of the class of systems under long-range interactions. We check existing results on the stability of the homogeneous situation and analyze numerical properties of the semi-Lagrangian time-split algorithm for solving the Vlasov equation. We also detail limitations due to finite resolution of the method.