On the numerical approximations of the periodic Schr\"odinger equation

TL;DR

This paper investigates the numerical approximation of the periodic Schr"odinger equation using finite difference schemes, identifying issues with high-frequency spurious solutions and proposing filtering and viscous methods to ensure uniform $L^4$ estimates.

Contribution

It introduces two methods—spectral filtering and viscous schemes—to achieve uniform $L^4$ space-time bounds in numerical solutions of the periodic Schr"odinger equation.

Findings

Blow-up in $L^4$ norm occurs with simple finite differences as mesh size decreases.

Spectral filtering of initial data restores uniform $L^4$ bounds.

Viscous scheme also achieves uniform $L^4$ estimates.

Abstract

We consider semidiscrete finite differences schemes for the periodic Scr\"odinger equation in dimension one. We analyze whether the space-time integrability properties observed by Bourgain in the continuous case are satisfied at the numerical level uniformly with respect to the mesh size. For the simplest finite differences scheme we show that, as mesh size tends to zero, the blow-up in the $L^4$ time-space norm occurs, a phenomenon due to the presence of numerical spurious high frequencies. To recover the uniformity of this property we introduce two methods: a spectral filtering of initial data and a viscous scheme. For both of them we prove a $L^4$ time-space estimate, uniform with respect to the mesh size. {Warning 2019}: This paper was submitted to M2AN in 2007 and it was assigned the number 2007-29. It passed a first review round ( three reviews :-) ) without decision ("It is only after the consideration of a thoroughly revised version of your manuscript and a new iteration with all the referees that I will be in position to make my final decision.") After completing the PhD, I tried to publish the papers resulting from the thesis and I didn't spent too much time on other related problems (the periodic case for example). It was only recently that I discovered some interest in the subject from other authors and I decided to upload it on arxiv.org. Use with caution as there is no revision of the text in the last twelve years.