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

2024/02/28 by Arnold, Anton, Körner, Jannis
#34E20 #65L11 #65M70 #81Q20 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2402.18406

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 [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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