Fast calculation of energy and mass preserving solutions of Schrödinger–Poisson systems on unbounded domains

This paper deals with the numerical solution of the time-dependent Schrödinger–Poisson system in the spherically symmetric case. Since the problem is posed on an unbounded domain one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of a so-called discrete transparent boundary condition (TBC) for a Crank–Nicolson-type predictor–corrector scheme for solving the Schrödinger–Poisson system. This scheme has the property of mass and energy conservation exactly on the discrete level. We propose different strategies for the discrete TBC and present an efficient implementation. Finally, a numerical example illustrate the findings and shows the comparison results between the different approaches.