The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip
1 CEA, DAM, DIF,
91297, Arpajon, France.
2 Department of Higher Mathematics at Faculty of Economics, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russia.
3 Department of Mathematical Modelling, National Research University Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russia.
Revised: 29 July 2013
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 L2-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.
Mathematics Subject Classification: 65M06 / 65M12 / 35Q40
Key words: The time-dependent Schrödinger equation / the Crank–Nicolson finite-difference scheme / the Strang splitting / approximate and discrete transparent boundary conditions / stability / tunnel effect
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