Abstract
This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid “turning points”. We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests.
Highlights
This paper deals with the design, error analysis, and numerical study of an asymptotically correct scheme for the numerical solution of the stationary Schrödinger equation in one dimensional scattering situations: ε2ψ (x) + a(x)ψ(x) = 0, x ∈ R, (1.1)where 0 < ε 1 is a very small parameter and a(x) a piecewise smooth, real function
We aim at recovering these fine structures of the solution, without using a fine spatial grid. To this end we shall develop a domain decomposition method (DDM) to separate the oscillatory and evanescent regions, as they require very different numerical approaches. This DDM allows to recover at the continuous level the exact analytical solution in a single sweep with appropriate interface conditions and a final scaling
This paper is concerned with a 1D Schrödinger scattering problem in the semi-classical limit, with the inflow given by plane waves
Summary
To this end we shall develop a (non-overlapping) domain decomposition method (DDM) to separate the oscillatory and evanescent regions, as they require very different numerical approaches This DDM allows to recover at the continuous level the exact analytical solution in a single sweep (against the direction of the incoming plane wave) with appropriate interface conditions and a final scaling. In the oscillatory region we shall use the second order WKB functions φ2os to transform (1.1) into a smoother problem that can be solved accurately and efficiently on a coarse grid.
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