Hamiltonian interpolation of splitting approximations for nonlinear PDEs

Erwan Faou (1, 2) , Benoît Grébert (3)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LMJL - Laboratoire de Mathématiques Jean Leray

We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni- formly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian equation whose exact solution coincides with the discrete flow at each time step over a long time depending on a non resonance condition satisfied by the stepsize. We introduce a class of modified splitting schemes fulfilling this condition at a high order and prove for them that the numerical flow and the continuous flow remain close over exponentially long time with respect to the step size. For stan- dard splitting or implicit-explicit scheme, such a backward error analysis result holds true on a time depending on a cut-off condition in the high frequencies (CFL condition). This analysis is valid in the case where the linear operator has a discrete (bounded domain) or continuous (the whole space) spectrum.

Mots clés: Hamiltonian interpolation; Backward error analysis; Splitting integrators; Nonlinear Schrödinger equation; Nonlinear wave equation; Long-time behavior

Domaines: Analyse numérique [math.NA]; Systèmes dynamiques [math.DS]

Erwan Faou, Benoît Grébert. Hamiltonian interpolation of splitting approximations for nonlinear PDEs. Foundations of Computational Mathematics, 2011, 11 (4), pp.381-415. ⟨10.1007/s10208-011-9094-4⟩. ⟨hal-00440982⟩