On the Crank-Nicolson scheme once again
type: Item
instance of: scholarly article
title: On the Crank-Nicolson scheme once again
zbMATH Open document ID: 1239.47035
published in: Journal of Evolution Equations
publication date: 2012-01-13T00:00:00Z
review text: Let \((\tau_j)_{j\in\mathbb{N}}\) be a sequence of strictly positive real numbers, and let \(A\) be the generator of a bounded analytic semigroup in a Banach space \(X\). Put \(A_n=\prod_{j=1}^n(I+\frac{1}{2} \tau_jA) (I-\frac{1}{2} \tau_jA)^{-1}\), and let \(x\in X\). Define the sequence \((x_n)_{n\in\mathbb{N}}\subset X\) by the Crank-Nicolson scheme: \(x _n = A_n x\). In this paper, it is proved that the Crank-Nicolson scheme is stable in the sense that \(\sup_{n\in\mathbb{N}}\| A_nx\|<\infty\). A serious flaw affects the main stability result. The author gives the correct formulation in an Erratum [\textit{J. A. van Casteren}, ``Erratum to: On the Crank-Nicolson scheme once again'', J. Evol. Equ. 11, No. 2, 477--483 (2011; Zbl 1239.47034)].
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34K30
zbMATH DE Number: 5997675
zbMATH Keywords: Crank-Nicolson scheme
zbMATH Keywords: bounded analytic semigroup
zbMATH Keywords: numerical methods
author: J. Van Casteren
MaRDI profile type: Publication
full work available at URL: s00028-010-0084-9
OpenAlex ID: W2021699159
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DOI: 10.1007/S00028-010-0084-9
description: scientific article; zbMATH DE number 5997675