  This paper considers the numerical treatment of the time-dependent
Gross-Pitaevskii equation. In order to conserve the time invariants of the
equation as accurately as possible, we propose a Crank-Nicolson-type time
discretization that is combined with a suitable generalized finite element
discretization in space. The space discretization is based on the technique of
Localized Orthogonal Decompositions (LOD) and allows to capture the time
invariants with an accuracy of order $\mathcal{O}(H^6)$ with respect to the
chosen mesh size $H$. This accuracy is preserved due to the conservation
properties of the time stepping method. Furthermore, we prove that the
resulting scheme approximates the exact solution in the $L^{\infty}(L^2)$-norm
with order $\mathcal{O}(\tau^2 + H^4)$, where $\tau$ denotes the step size. The
computational efficiency of the method is demonstrated in numerical experiments
for a benchmark problem with known exact solution.
