Section: New Results
Modified energy for split-step methods applied to the linear Schrödinger equation.
Participants : Arnaud Debussche, Erwan Faou.
In this work  we consider the linear Schrödinger equation and its discretization by split-step methods where the part corresponding to the (unbounded) 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 over arbitrarily long time. 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 arbitrarily long time. The analysis is valid in the case where the Schrödinger equation is set on a domain with periodic boundary condition, or on the whole space. This “backward error analysis" result is the first one given for the case of symplectic method applied to a Hamiltonian Partial differential equation (which is considered here as an infinite dimensional ordinary differential equation).