arXiv subjects

Jannis Körner

Publications and source records attributed to Jannis Körner.

5 recordsLinked to original sources

arXiv2025

WKB-based third order method for the highly oscillatory 1D stationary Schrödinger equation

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal., 2011], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

math.NA↗

arXiv2024

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter $\varepsilon$. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of $\varepsilon$ and the truncation order $N$. For any fixed $\varepsilon$, this allows to determine the optimal truncation order $N_{opt}$ which turns out to be proportional to $\varepsilon^{-1}$. When chosen this way, the resulting error of the optimally truncated WKB series behaves like $\mathcal{O}(\exp(-r/\varepsilon))$, with some parameter $r>0$. The theoretical results established in this paper are confirmed by several numerical examples.

math.NA↗

arXiv2023

High-order WKB-based Method For The 1D Stationary Schrödinger Equation In The Semi-classical Limit

We consider initial value problems for $\varepsilon^2\,φ''+a(x)\,φ=0$ in the highly oscillatory regime, i.e., with $a(x)>0$ and $0<\varepsilon\ll 1$. We discuss their efficient numerical integration on coarse grids, but still yielding accurate solutions. The $\mathcal{O}(h^2)$ one-step method from [2] is based on an analytic WKB-preprocessing of the equation. Here we extend this method to $\mathcal{O}(h^3)$ accuracy.

math.NA↗

arXiv2023

Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation

We discuss the numerical solution of initial value problems for $\varepsilon^2\,φ''+a(x)\,φ=0$ in the highly oscillatory regime, i.e., with $a(x)>0$ and $0<\varepsilon\ll 1$. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude $\mathcal{O}(\varepsilon^{N})$ where $N$ refers to the truncation order of the underlying asymptotic series. When the optimal truncation order $N_{opt}$ is chosen, the error behaves like $\mathcal{O}(\varepsilon^{-2}\exp(-c\varepsilon^{-1}))$ with some $c>0$.

math.NA↗

arXiv2021

WKB-based scheme with adaptive step size control for the Schrödinger equation in the highly oscillatory regime

This paper is concerned with an efficient numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Being a hybrid, analytical-numerical approach it does not have to resolve each oscillation, in contrast to standard schemes for ODEs. We build upon the WKB-based (named after the physicists Wentzel, Kramers, Brillouin) marching method from [2] and extend it in two ways: By comparing the $\mathcal{O}(h)$ and $\mathcal{O}(h^{2})$ methods from [2] we design an adaptive step size controller for the WKB method. While this WKB method is very efficient in the highly oscillatory regime, it cannot be used close to turning points. Hence, we introduce for such regions an automated methods switching, choosing between the WKB method for the oscillatory region and a standard Runge-Kutta-Fehlberg 4(5) method in smooth regions. A similar approach was proposed recently in [9, 4], however, only for an $\mathcal{O}(h)$-method. Hence, we compare our new strategy to their method on two examples (Airy function on the spatial interval $[0,\,10^{8}]$ with one turning point at $x=0$ and on a parabolic cylinder function having two turning points), and illustrate the advantages of the new approach w.r.t.\ accuracy and efficiency.

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