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

Peter A. Markowich; Paola Pietra; Carsten Pohl; Hans Peter Stimming

doi:10.1137/S0036142900381734

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

Peter A. Markowich^*, Paola Pietra, Carsten Pohl, Hans Peter Stimming

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

42 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 language	English (US)
Pages (from-to)	1281-1310
Number of pages	30
Journal	SIAM Journal on Numerical Analysis
Volume	40
Issue number	4
DOIs	https://doi.org/10.1137/S0036142900381734
State	Published - Sep 2002
Externally published	Yes

Keywords

Finite difference scheme
Schrödinger equation
Semiclassical limit
Wigner measure

ASJC Scopus subject areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

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10.1137/S0036142900381734

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title = "A Wigner-measure analysis of the Dufort-Frankel scheme for the Schr{\"o}dinger equation",

abstract = "We apply Wigner transform techniques to the analysis of the Dufort-Frankel difference scheme for the Schr{\"o}dinger 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 Schr{\"o}dinger equation.",

keywords = "Finite difference scheme, Schr{\"o}dinger equation, Semiclassical limit, Wigner measure",

author = "Markowich, {Peter A.} and Paola Pietra and Carsten Pohl and Stimming, {Hans Peter}",

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doi = "10.1137/S0036142900381734",

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T1 - A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrödinger equation

AU - Markowich, Peter A.

AU - Pietra, Paola

AU - Pohl, Carsten

AU - Stimming, Hans Peter

PY - 2002/9

Y1 - 2002/9

N2 - 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.

AB - 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.

KW - Finite difference scheme

KW - Schrödinger equation

KW - Semiclassical limit

KW - Wigner measure

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JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 4

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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