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High Order Efficient Splittings for the SemiclassicalTime–Dependent Schrödinger Equation

by S. Blanes and V. Gradinaru

(Report number 2019-36)

Abstract
Standard numerical schemes with time--step $h$ deteriorate (e.g. like $\varepsilon ^{-2} h^{2}$) in the presence of a small semiclassical parameter $\varepsilon $ in the time--dependent Schr\"{o}dinger equation. The recently introduced semiclassical splitting was shown to be of order $\mathcal{O}\left (\varepsilon h^{2}\right )$. We present now an algorithm that is of order $\mathcal{O}\left (\varepsilon h ^{7} + \varepsilon ^{2} h^{6} + \varepsilon ^{3} h^{4}\right )$ at the expense of roughly three times the computational effort of the semiclassical splitting and another that is of order $\mathcal{O} \left (\varepsilon h^{6} + \varepsilon ^{2} h^{4}\right )$ at the \emph{same} expense of the computational effort of the semiclassical splitting.

Keywords: semiclassical; time-dependent Schrödinger equation; splitting; wavepackets

@Techreport{BG19_840,
  author = {S. Blanes and V. Gradinaru},
  title = {High Order Efficient Splittings for the SemiclassicalTime–Dependent Schr\"odinger Equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-36.pdf },
  year = {2019}
}

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