Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

¹ RMAR & INRIA Rennes, Équipe IPSO, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cedex, France. francois.castella@univ-rennes1.fr
² INRIA Rennes, Équipe IPSO, Antenne de Bretagne de l'École Normale Supérieure de Cachan, Avenue Robert Schumann, 35170 Bruz, France. Guillaume.Dujardin@ens-cachan.org

Received: 10 September 2008

Abstract

In this paper, we study the linear Schrödinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends [G. Dujardin and E. Faou, Numer. Math. 97 (2004) 493–535], where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable.