  We compare two different numerical methods to integrate in time spatially
delocalized initial densities using the Schr\"odinger-Poisson equation system
as the evolution law. The basic equation is a nonlinear Schr\"odinger equation
with an auto-gravitating potential created by the wave function density itself.
The latter is determined as a solution of Poisson's equation modelling, e.g.,
non-relativistic gravity. For reasons of complexity, we treat a one-dimensional
version of the problem whose numerical integration is still challenging because
of the extreme long-range forces (being constant in the asymptotic limit). Both
of our methods, a Strang splitting scheme and a basis function approach using
B-splines, are compared in numerical convergence and effectivity. Overall, our
Strang-splitting evolution compares favourably with the B-spline method. In
particular, by using an adaptive time-stepper rather large one-dimensional
boxes can be treated. These results give hope for extensions to two spatial
dimensions for not too small boxes and large evolution times necessary for
describing, for instance, dark matter formation over cosmologically relevant
scales.
