An averaging technique for highly-oscillatory Hamiltonian problems

François Castella (1, 2) , Philippe Chartier (1) , Erwan Faou (1, 2)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we are concerned with the numerical solution of highly-oscillatory Hamiltonian systems with a stiff linear part. We construct an averaged system whose solution remains close to the exact one over bounded time intervals, possesses the same adiabatic and Hamiltonian invariants as the original system, and is non-stiff. We then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time-averages developped in a previous paper. Eventually, we demonstrate the efficiency of our approach on two test problems with one or several frequencies.

Mots clés: filtering; highly-oscillatory; averaging; adiabatic invariance; filtering.

Domaines: Equations aux dérivées partielles [math.AP]

François Castella, Philippe Chartier, Erwan Faou. An averaging technique for highly-oscillatory Hamiltonian problems. SIAM Journal on Numerical Analysis, 2009, 47 (4), pp.2808--2837. ⟨10.1137/080715974⟩. ⟨hal-00777047⟩