A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation

Peter Markowich*, Paola Pietra, Carsten Pohl, Hans Peter Stimming

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

We apply Wigner transform techniques to the analysis of the Dufort-Frankel difference scheme for the Schrödinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort-Frankel scheme to other difference schemes for the Schrödinger equation.

Original languageEnglish (US)
Pages (from-to)1281-1310
Number of pages30
JournalSIAM Journal on Numerical Analysis
Volume40
Issue number4
DOIs
StatePublished - Sep 1 2002

Keywords

  • Finite difference scheme
  • Schrödinger equation
  • Semiclassical limit
  • Wigner measure

ASJC Scopus subject areas

  • Numerical Analysis

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