WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation
Numerical Analysis
2025-04-29 v2 Numerical Analysis
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.
Cite
@article{arxiv.2402.18406,
title = {WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation},
author = {Anton Arnold and Jannis Körner},
journal= {arXiv preprint arXiv:2402.18406},
year = {2025}
}
Comments
42 pages, 10 figures