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

Arnaud Debussche, Erwan Faou
2009 SIAM Journal on Numerical Analysis  
We consider the linear Schrödinger equation and its discretization by splitstep 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
more » ... equence, we give uniform regularity estimates for the numerical solution over arbitrary long time. MSC numbers: 65P10, 37M15
doi:10.1137/080744578 fatcat:ikregeybafdqzbzfnhuyw26tsa