Stability and error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions

A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schrödinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of z z -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green’s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.