  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.
