Stability of the Peaceman-Rachford approximation
type: Item
instance of: scholarly article
title: Stability of the Peaceman-Rachford approximation
zbMATH Open document ID: 0920.47021
author: Michelle Schatzman
published in: Journal of Functional Analysis
publication date: 1999-09-12T00:00:00Z
review text: Let \(A\) and \(B\) be selfadjoint bounded from below operators in a Hilbert space. The author investigates stability of approximation (in the operator norm) of \(e^{-t(A+B)}\) by the Peaceman-Rachford formula \[ p(t/2n)^n \to e^{-t(A+B)}, \] where \(p(t) = (1+tA)^{-1}(1-tB)(1+tB)^{-1}(1-tA)\). The approximation is stable if \(\| p(t)\| \leq 1+O(t)\). Sufficient conditions for stability are given; they involve conditions on the commutators of \(\sqrt A\) and \(\sqrt B\).
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1270901
zbMATH Keywords: stability of approximation
zbMATH Keywords: Peaceman-Rachford formula
zbMATH Keywords: commutators
MaRDI profile type: Publication
full work available at URL: jfan.1998.3311
OpenAlex ID: W2061287893
cites work: Note on product formulas for operator semigroups
cites work: Q4838252
cites work: Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées
cites work: An estimate of the Kac transfer operator
cites work: On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods
cites work: Q3910982
cites work: Q3679959
cites work: Q3736112
cites work: Q3875260
cites work: Q3126807
cites work: The Numerical Solution of Parabolic and Elliptic Differential Equations
cites work: On the Product of Semi-Groups of Operators
DOI: 10.1006/JFAN.1998.3311
description: scientific article; zbMATH DE number 1270901