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March  2008, 1(1): 101-125. doi: 10.3934/krm.2008.1.101

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

Maike Schulte 1, and  Anton Arnold 2,

1. 

Institut für Angewandte und Numerische Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
2. 

Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, 1040 Wien, Australia

Received  September 2007 Revised  October 2007 Published  February 2008

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: finite difference schemes, transparent boundary conditions, compact discretization schemes., Schrödinger equation.
Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35Q4.
Citation: Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme. Kinetic & Related Models, 2008, 1 (1) : 101-125. doi: 10.3934/krm.2008.1.101
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