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A detailed exposition of highly efficient and accurate method for thepropagation of the time-dependent Schr\"odinger equation is presented. Themethod is readily generalized to solve an arbitrary set of ODE's. Thepropagation is based on a global approach, in which large time-intervals aretreated as a whole, replacing the local considerations of the commonpropagators. 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 thispaper, a thorough presentation of the basic principles of the propagator isgiven. We give also a special emphasis on the details of the numericalimplementation of the method. For the first time, we present the applicationfor a non-Hermitian problem by a numerical example of a one-dimensional atomunder the influence of an intense laser field. The efficiency of the method isdemonstrated by a comparison with the common Runge-Kutta approach.

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