[Back]

A. Arnold, N. Ben Abdallah, C. Negulescu:
"WKB-based schemes for the Schr\"odinger equation in the semi-classical limit";
in: "ASC Report 04/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

An efficient and accurate numerical method is presented for the solution of
highly oscillatory differential equations. While standard methods would require
a very fine grid to resolve the oscillations, the presented approach uses first an
analytic (second order) WKB-type transformation, which filters out the dominant
oscillations. The resulting ODE is much smoother and can hence be discretized
on a much coarser grid, with significantly reduced numerical costs.
In many practically relevant examples, the method is even asymptotically correct
w.r.t. the small parameter " that identifies the oscillation wave length. Indeed,
in these cases, the error then vanishes for " → 0, even on a fixed spatial mesh.
Applications to the stationary Schr¨odinger equation are presented.

Schr¨odinger equation; highly oscillating wave functions; epsilon-independent

http://www.asc.tuwien.ac.at/preprint/2009/asc04x2009.pdf