Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations

In the paper, we propose a family of high order compact ADI (HOC–ADI) scheme for multi-dimensional Schrödinger equations. In a specific case, it is of sixth-order accuracy in space. To cut down the computational labor, we adopt ADI strategy in the time direction. Moreover, for nonlinear problem, the second-order standard Strang splitting skill is used. The conservation properties and stability are analyzed for the proposed scheme. Numerical results in 2d and 3d are reported to demonstrate the new spectral-like resolution. Numerical results suggest that the proposed scheme is very efficient and accurate.

► The high order compact (HOC) method is almost as accurate as spectral method with much less time.
► The HOC method is combined with the ADI method with greatly diminishing resources.
► The method is extended to nonlinear problem by introducing the splitting method.
► Excellent conservation properties are presented by the method.
► A great deal of persuasive numerical results are obtained.