Efficient algorithms for Schrödinger equations with concentrated potential

We propose an efficient family of algorithms for the approximation of non linear Schrödinger equations with concentrated potentials by using Runge-Kutta based convolution quadrature. The algorithms are based on a novel integral representation of the convolution quadrature weights and a special quadrature for it. The resulting method allows for high order and its performance in terms of memory and computational cost allows for long time simulations. Furthermore, the new algorithm is easy to implement, since it is mosaic-free, i.e. it does not require any sophisticated memory management. The algorithm generalizes ideas used recently by the authors to approximate time-fractional differential equations. This is a joint work with Lehel Banjai from Heriot-Watt University, Edinburgh, UK.