| Abstract: |
| 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. We shall discuss the efficient numerical integration of ODEs of the form $\epsilon^2 u + a(x) u = 0$ for $0 < \epsilon \ll 1$ 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 $a(x) < 0$ ). 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 $h = O(\epsilon^{7/12})$. |
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