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

Bernard Ducomet; Alexander Zlotnik; Ilya Zlotnik

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 6, page 1681-1699
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Ducomet, Bernard, Zlotnik, Alexander, and Zlotnik, Ilya. "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 - Modélisation Mathématique et Analyse Numérique 48.6 (2014): 1681-1699. <http://eudml.org/doc/273296>.

@article{Ducomet2014,
abstract = {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.},
author = {Ducomet, Bernard, Zlotnik, Alexander, Zlotnik, Ilya},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {the time-dependent Schrödinger equation; the Crank–Nicolson finite-difference scheme; the strang splitting; approximate and discrete transparent boundary conditions; stability; tunnel effect; time-dependent Schrödinger equation; Crank-Nicolson finite-difference scheme; strang splitting; initial boundary value problem; algorithm; fast Fourier transform; numerical result; error analysis},
language = {eng},
number = {6},
pages = {1681-1699},
publisher = {EDP-Sciences},
title = {The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip},
url = {http://eudml.org/doc/273296},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Ducomet, Bernard
AU - Zlotnik, Alexander
AU - Zlotnik, Ilya
TI - The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 6
SP - 1681
EP - 1699
AB - 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.
LA - eng
KW - the time-dependent Schrödinger equation; the Crank–Nicolson finite-difference scheme; the strang splitting; approximate and discrete transparent boundary conditions; stability; tunnel effect; time-dependent Schrödinger equation; Crank-Nicolson finite-difference scheme; strang splitting; initial boundary value problem; algorithm; fast Fourier transform; numerical result; error analysis
UR - http://eudml.org/doc/273296
ER -

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