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Working paper

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

arxiv.org. math. Cornell University, 2013. No. arxiv: 1303.3471.
Ducomet B., Zlotnik A., Zlotnik I. A.
  We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation on a semi-infi nite strip. For the Crank-Nicolson fi nite-diff erence 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 uniform in time L2-stability is proved. Due to the splitting, an e ffective direct algorithm using FFT is developed to implement the method with the discrete TBC for general potential. Numerical results on the tunnel e ffect for rectangular barriers are included together with the related practical error analysis.