AMaSiS 2018 Workshop: Abstracts

A hybrid WKB-based method for Schrödinger scattering problems in the semi-classical limit

Anton Arnold

Technische Universität Wien, Institute for Analysis and Scientific Computing

We are concerned with 1D scattering problems related to quantum transport in (tunneling) diodes. The problem includes both oscillatory and evanescent regimes, partly including turning points.

1. WKB-scheme

We shall discuss the efficient numerical integration of ODEs of the form ${\epsilon }^2{u}^{\mathrm{\prime \prime }}+a(x)u=0$ for on coarse grids, but still yielding accurate solutions. In particular we study the numerical coupling of the highly oscillatory regime (i.e. for given $a(x)>0$) with evanescent regions (i.e. for ). In the oscillatory case we use a marching method that is based on an analytic WKB-preprocessing of the equation. And in the evanescent case we use a FEM with WKB-ansatz functions.

We present a full convergence analysis of the coupled method, showing that the error is uniform in epsilon and second order w.r.t. $h$, when $h=O({\epsilon }^{1/2})$. We illustrate the results with numerical examples for scattering problems for a quantum-tunnelling structure.

The main challenge when including a turning point is that the solution gets unbounded there as $\epsilon \to 0$. Still one can obtain epsilon-uniform convergence, when $h=O({\epsilon }^{7/12})$.

Acknowledgments: The author was supported by the FWF-doctoral school “Dissipation and dispersion in non-linear partial differential equations” and the FWF-funded SFB F-65