Improvement on stability and convergence of ADI schemes
type: Item
instance of: scholarly article
title: Improvement on stability and convergence of ADI schemes
zbMATH Open document ID: 0931.65095
DOI: 10.1007/BF02459276
author: Aijie Cheng
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 2000-02-15T00:00:00Z
review text: Two alternating direction implicit (ADI) schemes namely, the Peaceman-Rachford scheme and Douglas scheme for the two-dimensional quasilinear parabolic problem are studied. In the literature, stability and convergence have been analysed with the Fourier method, which can not be extended beyond the model problem with constant coefficients. Additionally, the \(L^{2}\) energy method has been introduced to analyse the case of non-constant coefficients, however, the conclusions are too weak and incomplete because of the so-called ``equivalence between the \(L^{2}\) norm and the \(H^{1}\) semi-norm''. In this paper the author tries to improve these conclusions by the \(H^{1}\) energy estimating method. The principal results are that both of the two ADI schemes are absolutely stable and converge to the exact solution with error estimations \(O (\triangle t^{2} +h^{2}) \) in discrete \(H^{1}\) norm.
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 1346284
zbMATH Keywords: alternating direction implicit schemes
zbMATH Keywords: Douglas scheme
zbMATH Keywords: Peaceman-Rachford scheme
zbMATH Keywords: variable coefficient
zbMATH Keywords: \(H^{1} \) energy estimating method
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: quasilinear parabolic problem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02459276
OpenAlex ID: W2076688751
description: scientific article; zbMATH DE number 1346284