WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation
<|reference_start|>WKB-based third order method for the highly oscillatory 1D stationary Schr\"odinger equation: 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 [2], 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.<|reference_end|>
@article{arnold2024wkb-based,
    title={WKB-based third order method for the highly oscillatory 1D stationary
  Schr\"odinger equation},
    author={Anton Arnold, Jannis K"orner},
    journal={arXiv preprint arXiv:2402.18406},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.18406},
    primaryClass={math.NA cs.NA}
}