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On the Crank-Nicolson scheme once again

On the Crank-Nicolson scheme once again Let $${\left(\tau_j\right)_{j\in\mathbb{N}}}$$ 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 $${A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}$$ , and let $${x\in X}$$ . Define the sequence $${\left(x_n\right)_{n\in\mathbb{N}}\subset X}$$ 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 $${\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}$$ . Some convergence results are also given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

On the Crank-Nicolson scheme once again

Casteren, Jan
Journal of Evolution Equations , Volume 11 (2) – Jun 1, 2011
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References (14)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-010-0084-9
Publisher site
See Article on Publisher Site

Abstract

Let $${\left(\tau_j\right)_{j\in\mathbb{N}}}$$ 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 $${A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}$$ , and let $${x\in X}$$ . Define the sequence $${\left(x_n\right)_{n\in\mathbb{N}}\subset X}$$ 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 $${\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}$$ . Some convergence results are also given.

Journal

Journal of Evolution Equations – Springer Journals

Published: Jun 1, 2011

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