Discrete Schrödinger-Poisson systems preserving energy and mass

Abstract

A discrete predictor-corrector Schrödinger-Poisson system is proposed which has the property of mass and energy conservation exactly on the discrete level. The discretization is based on the Cranck-Nicholson scheme, which preserves these invariants of the Schrödinger-Poisson system, but involves the solution of nonlinear equations at each time step. A modified linearized scheme is proposed where conservation is achieved by introducing a phase modulation in the corrector step.