"Numerical simulation of nonlinear Schrödinger equations"; Prof. Dr. Daniel Peterseim (Universität Augsburg)

ABSTARCT: This talk reviews numerical methods for the simulation of Bose-Einstein-Condensates modeled by nonlinear Schrödinger (Gross-Pitaevskii) equations with possibly rough (highly oscillatory, disordered) potentials. The first part addresses the computation of stationary states. Among the methodological and mathematical novelties for the nonlinear Schrödinger eigenvalue problem are globally convergent and energy-diminishing gradient flows as well as shift-accelerated nonlinear inverse iterations. The second part concerns the dynamics and the numerical analysis of time-stepping schemes in the presence of disorder potentials. Under low regularity assumptions, that are compatible with rough potentials, we prove convergence with rates for the mass- and energy conserving variant of the Crank-Nicolson time discretization scheme due to Sanz-Serna. A series of numerical experiments demonstrates the computational efficiency of all methods and their ability to capture interesting physical phenomena such as Anderson localization and quantum phase transitions.