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

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.