Semi-global approach for propagation of the time-dependent Schr\"odinger equation for time-dependent and nonlinear problems

A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schr\"odinger equation is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge-Kutta approach.