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

TL;DR

This paper introduces a modified energy concept for split-step methods applied to the linear Schrödinger equation, demonstrating energy preservation and long-term regularity of numerical solutions.

Contribution

It establishes a modified energy framework for split-step methods with midpoint rule approximation, ensuring energy conservation and stability over long times.

Findings

Numerical solution matches a modified PDE at each step.

Modified energy is conserved by the scheme.

Long-term regularity estimates are provided.

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