The Schrödinger equation with a time-dependent potential occurs in a wide range of applications in theoretical chemistry, quantum physics and quantum computing. In this talk I will discuss a variety of Magnus expansion based schemes that have been found to be highly effective for numerically solving these equations since the pioneering work of Tal Ezer and Kosloff in the early 90s. Recent developments in the field focus on approximation of the exponential of the Magnus expansion via exponential splittings including some asymptotic splittings and commutator-free splittings that are designed specifically for this task.
I will also present a very recently developed methodology for the case of laser-matter interaction. This methodology allows us to extend any fourth-order scheme for Schrödinger equation with time-independent potential to a fourth-order method for Schrödinger equation with laser potential with little to no additional cost. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.