Modified energy for split-step methods applied to the linear Schrödinger equation

Arnaud Debussche (1) , Erwan Faou (2)

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

We consider the linear Schrödinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time

Mots clés: Schrödinger equation; Splitting integrators; Long-time behavior; Backward error analysis

Domaines: Analyse numérique [math.NA]

Arnaud Debussche, Erwan Faou. Modified energy for split-step methods applied to the linear Schrödinger equation. SIAM Journal on Numerical Analysis, 2009, 47 (5), pp.3705-3719. ⟨hal-00348221⟩