  Based on recent ideas, stemming from the use of bubbles, we discuss an
algorithm for the numerical simulation of the cubic nonlinear Schr{\"o}dinger
equation with harmonic potential in any dimension, which could be easily
extended to other polynomial nonlinearities. For the linear part of the
equation, the algorithm consists in discretizing the initial function as a sum
of modulated complex functions, each one having its own set of parameters, and
then updating the parameters exactly so that the modulated function remains a
solution to the equation. When cubic interactions are introduced, the
Dirac-Frenkel-MacLachlan principle is used to approximate the time evolution of
parameters. We then obtain a grid-free algorithm in any dimension, and it is
compared to a spectral method on numerical examples.
