Kinetic and Related Models (KRM)

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

Pages: 151 - 179, Volume 2, Issue 1, March 2009      doi:10.3934/krm.2009.2.151

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Bernard Ducomet - DPTA/Service de Physique Nucléaire, CEA, DAM, DIF, F-91297 Arpajon, France (email)
Alexander Zlotnik - Department of Applied Mathematics, Russian State Social University, W. Pieck 4, 129226 Moscow, Russian Federation (email)
Ilya Zlotnik - Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation (email)

Abstract: 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:  Stability, finite-difference schemes, finite element method, approximate and discrete transparent boundary conditions, the time-dependent Schrödinger equation, discrete convolution operator.
Mathematics Subject Classification:  Primary: 65M06, 65M12, 65M60; Secondary: 35Q40.

Received: November 2008;      Revised: January 2009;      Available Online: January 2009.