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We present a novel numerical method and algorithm for the solution of the 3Daxially symmetric time-dependent Schr\"odinger equation in cylindricalcoordinates, involving singular Coulomb potential terms besides a smoothtime-dependent potential. We use fourth order finite difference real spacediscretization, with special formulae for the arising Neumann and Robinboundary conditions along the symmetry axis. Our propagation algorithm is basedon merging the method of the split-operator approximation of the exponentialoperator with the implicit equations of second order cylindrical 2DCrank-Nicolson scheme. We call this method hybrid splitting scheme because itinherits both the speed of the split step finite difference schemes and therobustness of the full Crank-Nicolson scheme. Based on a thorough erroranalysis, we verified both the fourth order accuracy of the spatialdiscretization in the optimal spatial step size range, and the fourth orderscaling with the time step in the case of proper high order expressions of thesplit-operator. We demonstrate the performance and high accuracy of our hybridsplitting scheme by simulating optical tunneling from a hydrogen atom due to afew-cycle laser pulse with linear polarization.

Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential