  The time-dependent Gross-Pitaevksii equation (GPE) is a nonlinear
Schr\"odinger equation which is used in quantum physics to model the dynamics
of Bose-Einstein condensates. In this work we consider numerical approximations
of the GPE based on a multiscale approach known as the localized orthogonal
decomposition. Combined with an energy preserving time integrator one derives a
method which is of high order in space and time under mild regularity
assumptions. In previous work, the method has been shown to be numerically very
efficient compared to first order Lagrange FEM. In this paper, we further
investigate the performance of the method and compare it with higher order
Lagrange FEM. For rough problems we observe that the novel method performs very
efficient and retains its high order, while the classical methods can only
compete well for smooth problems.
