Modified Energy for Split-Step Methods Applied to the Linear Schrödinger Equation

Authors:
Arnaud Debussche;Erwan Faou
Affiliations:
Arnaud.Debussche@bretagne.ens-cachan.fr and Erwan.Faou@inria.fr;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2009

Citing 0
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Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

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Abstract

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 arbitrarily long time.