Averaging of highly-oscillatory transport equations

Philippe Chartier (1, 2) , Nicolas Crouseilles (1, 2) , Mohammed Lemou (1, 3, 2) , Florian Méhats (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 MINGUS - Multi-scale numerical geometric schemes
3 CNRS - Centre National de la Recherche Scientifique

In this paper, we develop a new strategy aimed at obtaining high-order asymp-totic 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.

Mots clés: normal form; formal series; averaging; transport equation; highly-oscillatory; Vlasov equation; regime; strong magnetic field; Mathematics Subject Classification (2010): 34C29; 82B40; 35Q83

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Physique des plasmas [physics.plasm-ph]

Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic and Related Models , 2020, 13 (6), pp.1107-1133. ⟨10.3934/krm.2020039⟩. ⟨hal-01396685⟩