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Building upon the ideas from the article (Arnold et al. SIAM J. Numer. Anal. <jats:bold>49<\/jats:bold>, 1436\u20131460, 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.<\/jats:p>","DOI":"10.1007\/s10444-025-10234-y","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T09:42:45Z","timestamp":1747302165000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["WKB-based third order method for the highly oscillatory 1D stationary Schr\u00f6dinger equation"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9923-2888","authenticated-orcid":false,"given":"Anton","family":"Arnold","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jannis","family":"K\u00f6rner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,15]]},"reference":[{"key":"10234_CR1","unstructured":"Advanpix LLC. 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