Geometric integrators for piecewise smooth Hamiltonian systems

Philippe Chartier (1) , Erwan Faou (1)

1 IPSO - Invariant Preserving SOlvers

In this paper, we consider C 1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411-418], and we prove it is convergent, and that it preserves the energy and the volume.

Mots clés: Hamiltonian systems; symplecticity; volume-preservation; energy-preservation; B-splines; weak order

Domaines: Analyse numérique [math.NA]

Philippe Chartier, Erwan Faou. Geometric integrators for piecewise smooth Hamiltonian systems. ESAIM: Mathematical Modelling and Numerical Analysis, 2008, 42 (2), pp.223-241. ⟨10.1051/m2an:2008006⟩. ⟨hal-00686857⟩