Geodesic
WKB-based schemes for two-band Schr\"odinger equations in the highly oscillatory regime
Nanosistemy: fizika, himiâ, matematika, Tome 2 (2011) no. 3, pp. 7-28.

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An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations in one spatial dimension. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic WKB-type transformation, which filters out the dominant oscillations. The resulting ODE-system is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs. Here we are concerned with stationary two-band Schrödinger equations employed in quantum transport applications. We focus on the Kane-model and the two band - model. The accuracy of the presented method is illustrated on a numerical example.
Keywords: Schrödinger equation, Kane-model, two-band -model, highly oscillating wave functions, higher order WKB-approximation, asymptotically correct finite difference scheme. 
@article{NANO_2011_2_3_a0,
     author = {J. Geier and A. Arnold},
     title = {WKB-based schemes for two-band {Schr\"odinger} equations in the highly oscillatory regime},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
     pages = {7--28},
     publisher = {mathdoc},
     volume = {2},
     number = {3},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2011_2_3_a0/}
}
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J. Geier; A. Arnold. WKB-based schemes for two-band Schr\"odinger equations in the highly oscillatory regime. Nanosistemy: fizika, himiâ, matematika, Tome 2 (2011) no. 3, pp. 7-28. http://geodesic.mathdoc.fr/item/NANO_2011_2_3_a0/