On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

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March 2009, 2(1): 151-179. doi: 10.3934/krm.2009.2.151

On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Bernard Ducomet ^1, , Alexander Zlotnik ^2, and Ilya Zlotnik ^3,

1.	DPTA/Service de Physique Nucléaire, CEA, DAM, DIF, F-91297 Arpajon, France
2.	Department of Applied Mathematics, Russian State Social University, W. Pieck 4, 129226 Moscow, Russian Federation
3.	Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation

Received November 2008 Revised January 2009 Published January 2009

Abstract
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We consider a 1D Schrödinger equation with variable coefficients on the half-axis. We study a family of two-level symmetric finite-difference schemes with a three-point parameter dependent averaging in space. This family includes a number of particular schemes. The schemes are coupled to an approximate transparent boundary condition (TBC). We prove two stability bounds with respect to initial data and a free term in the main equation, under suitable conditions on an operator of the approximate TBC. We also consider the family of schemes on an infinite mesh in space. We derive and analyze the discrete TBC allowing to restrict these schemes to a finite mesh and prove the stability conditions for it. Numerical examples are also included.

Keywords: finite-difference schemes, the time-dependent Schrödinger equation, finite element method, approximate and discrete transparent boundary conditions, Stability, discrete convolution operator..

Mathematics Subject Classification: Primary: 65M06, 65M12, 65M60; Secondary: 35Q4.

Citation: Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

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