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 Schršodinger equation are presented.

Schršodinger equation; highly oscillating wave functions; epsilon-independent

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

Created from the Publication Database of the Vienna University of Technology.