Kinetic and Related Models (KRM)

Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme

Pages: 101 - 125, Volume 1, Issue 1, March 2008      doi:10.3934/krm.2008.1.101

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Maike Schulte - Institut für Angewandte und Numerische Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany (email)
Anton Arnold - Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, 1040 Wien, Australia (email)

Abstract: We consider the two-dimensional time-dependent Schrödinger equation with the new compact nine-point scheme in space and the Crank-Nicolson difference scheme in time. For the resulting difference equation we derive discrete transparent boundary conditions in order to get highly accurate solutions for open boundary problems. Numerical experiments illustrate the perfect absorption of outgoing wave independently of their impact angle at the boundary. Finally, we apply inhomogeneous discrete transparent boundary conditions to the transient simulation of quantum waveguides.

Keywords:  Schrödinger equation, transparent boundary conditions, finite difference schemes, compact discretization schemes.
Mathematics Subject Classification:  Primary: 65M06, 65M12; Secondary: 35Q40.

Received: September 2007;      Revised: October 2007;      Available Online: February 2008.