On a Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) together with the error estimate $O(\tau^2+|h|^4)$ are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip and solving of tridiagonal systems in its main direction is developed to implement the splitting method for general potential. We also engage the Richardson extrapolation in time to increase the error order with respect to time step and get the method of higher order both in space and time. Numerical results on the tunnel effect for smooth and discontinuous rectangular barriers are included together with the careful practical error analysis on refining meshes.