Patrick Henning
In this talk we consider the numerical treatment of nonlinear Schrödinger equations as they appear for example in quantum physics and fluid dynamics. We give numerical examples that demonstrate the influence of the discrete energy on the accuracy of numerical approximations and that a spurious energy can create artificial phenomena such as drifting particles. In order to conserve the exact energy 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 general time invariants with a 6th order accuracy with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. The computational efficiency of the method is demonstrated for a numerical benchmark problem with known exact solution, which is however not solvable with traditional methods on long time scales.