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

IF 2.1 3区数学 Q2 MATHEMATICS, APPLIED

Advances in Computational Mathematics Pub Date : 2025-05-15 DOI:10.1007/s10444-025-10234-y

Anton Arnold, Jannis Körner

引用次数: 0

Abstract

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 the article (Arnold et al. SIAM J. Numer. Anal. 49, 1436–1460, 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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基于wkb的三阶方法求解高振荡一维平稳Schrödinger方程

本文介绍了一种求解高振荡状态下一维稳态Schrödinger方程的高效高阶数值方法。基于文章中的观点（Arnold et al.）。SIAM J. number。在论文（Anal. 49, 1436-1460, 2011）中，我们首先解析地将给定方程转换为更平滑（即振荡较小）的方程。通过对在解的皮卡德近似中出现的几个（迭代）振荡积分进行足够精确的正交，我们得到了一种步长为三阶的单步方法。通过算例说明了该方法的准确性和有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Computational Mathematics 数学-应用数学

CiteScore

3.00

自引率

5.90%

发文量

审稿时长

3 months

期刊介绍： Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis. This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.