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

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

2009-01-09

arxiv.org/abs/0901.1190v1

Mathematics

Numerical Analysis

Scientific paper

We consider the linear Schr\"odinger 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

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