SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Finite-difference schemes with splitting and discrete TBCs for the 2D Schrödinger equation in a strip

A. Zlotnik

Abstract
The splitting technique is widely used to simplify solving of the TD Schrödinger and related equations [1] which are so important in various fields. We apply the Strang-type splitting in potential to the Crank-Nicolson and Numerov-Crank-Nicolson schemes for the 2D TD Schrödinger in a strip with the discrete transparent boundary conditions (DTBC). We prove the uniform in time $L^2$-stability and the uniqueness of solutions.
We also need to study the splitting schemes on an infinite mesh in the strip and derive the uniform in time $L^2$-stability and the mass conservation law. The DTBC operator is written in terms of the discrete convolution in time and the discrete Fourier expansion in direction $y$ perpendicular to the strip. Effective direct algorithms using FFT in $y$ are developed to implement the schemes for general potential. Promising numerical results on the tunnel effect for some barriers and the practical error analysis in $C$ and $L^2$ norms are given.

Bibliography
[1] C. Lubich, From quantum to classical molecular dynamics. Reduced models and numerical analysis, EMS, Zürich, 2008.
[2] B. Ducomet, A. Zlotnik and I. Zlotnik, The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip, submitted, 2013, http://arxiv.org/abs/1303.3471.
[3] A. Zlotnik and A. Romanova, A Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip, submitted, 2013.

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