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arXiv · 2402.18406

WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation

Anton Arnold·Jannis Körner
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Abstract

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger 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.

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Anton Arnold, Jannis Körner. 2024-02-28. WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation. https://arxiv.org/abs/2402.18406

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