Averaging of highly-oscillatory transport equations

TL;DR

This paper introduces a novel high-order averaging method for highly-oscillatory transport equations, enabling accurate asymptotic modeling and exact solution recovery, demonstrated on Vlasov equations with magnetic fields.

Contribution

The paper develops a new averaging strategy for transport equations that achieves high-order accuracy and allows exact solution reconstruction, extending previous models and deriving new ones.

Findings

Successfully applied to Vlasov equations with magnetic fields

Re-derived existing asymptotic models using the new method

Produced new high-order asymptotic models for complex fields

Abstract

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature and re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.