The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

ESAIM: Mathematical Modelling and Numerical Analysis. 2014. Vol. 48. No. 6. P. 1681–1699.

Ducomet Bernard, Zlotnik Alexander, Zlotnik Ilya

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.

Keywords: устойчивость, stability, нестационарное уравнение Шредингера, The time-dependent Schrödinger equation, approximate and discrete transparent boundary conditions, приближенные и дискретные прозрачные граничные условия, туннельный эффект, the Crank-Nicolson finite-difference scheme, The Strang splitting, разностная схема Кранка-Никольсон, расщепление Стренга, tunnel effect

Publication based on the results of: Численные методы решения некоторых задач математической физики и газовой динамики (2012)