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Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Published online by Cambridge University Press:  09 September 2015

Emil Kieri ,
Gunilla Kreiss  and
Olof Runborg
Emil Kieri*
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden
Gunilla Kreiss
Affiliation:
Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden
Olof Runborg
Affiliation:
Department of Mathematics and Swedish e-Science Research Center (SeRC), KTH, Sweden
*
*Corresponding author. Email: [email protected] (E. Kieri), [email protected] (G. Kreiss), [email protected] (O. Runborg)
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Abstract

In the semiclassical regime, solutions to the time-dependent Schrödinger equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

Keywords

35Q4141A6065M06Gaussian beamssemiclassical Schrödinger equationhybrid methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 6 , December 2015 , pp. 687 - 714
Copyright
Copyright © Global-Science Press 2015 

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