| Contents |
| Plane wave numerical stability for cubic Schr\"odinger equation has
already been studied in the literature when integrating in time with
a first-order splitting and with some second-order implicit
methods. On the other hand, it has recently been observed that
exponential Lawson methods can be an efficient tool to integrate
this equation. In particular, when this type of methods are based
on explicit Runge-Kutta ones, after projecting on one invariant (norm),
the numerical solution arrives at another invariant (momentum) for
many solutions which include plane waves. As the projection is
very cheap, some times these methods are competitive against
splitting ones, which have also been proved to very efficient when
intetrating this equation.
In this talk, we will show the results of an exhaustive analysis on
plane wave numerical stability when integrating with the following
second-order explicit exponential methods: Strang splitting and
Lawson methods based on a one-parameter family of
2-stage Runge-Kutta ones. For the latter, we will consider the
projected and unprojected version onto the norm. We will show
stability regions and numerical experiments which corroborate
theoretical results. |
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