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arXiv · 0901.1190

Modified energy for split-step methods applied to the linear Schr\"odinger equation

Arnaud Debussche·Erwan Faou
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Abstract

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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Arnaud Debussche, Erwan Faou. 2009-01-09. Modified energy for split-step methods applied to the linear Schr\"odinger equation. https://arxiv.org/abs/0901.1190

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