On stability of two-layer explicit schemes
type: Item
instance of: scholarly article
zbMATH Open document ID: 0970.65097
author: P. V. Banushkina
author: Y. M. Laevsky
publication date: 2001-06-10T00:00:00Z
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1603285
zbMATH Keywords: compound explicit scheme
zbMATH Keywords: two-layer scheme
zbMATH Keywords: stability
zbMATH Keywords: convergence
MaRDI profile type: Publication
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title: On stability of two-layer explicit schemes
published in: Sibirskiĭ Zhurnal Vychislitel'noĭ Matematiki
review text: The authors continue in studying compound explicit schemes [see \textit{Yu. M. Laevskij} and \textit{P. Banushkina}, Sib. J. 3, No.~2, 165-180 (2000; Zbl 0956.65075)] wherein the authors investigated the questions of convergence and stability of the schemes under consideration.NEWLINENEWLINENEWLINEThe aim of the article under review is to expose conditions for the schemes which guarantee stability with respect to the right-hand of the scheme.NEWLINENEWLINENEWLINEThe following two-layer canonical scheme is considered: NEWLINE\[NEWLINE B\frac{u^{n+1} - u^n}{\Delta t} + Au^n = \varphi^n, NEWLINE\]NEWLINE where \(A\: H\to H\), \(B\: H\to H\) are matrix operators in a Hilbert space~\(H\) of the form NEWLINE\[NEWLINE A = \left( \begin{matrix} A_{11} & A_{12}\\ A_{12}^T & A_{22} \end{matrix} \right), \quad B = \left( \begin{matrix} I_{1} & 0_{12}\\ Q^{-1}RA^T_{12} & Q^{-1} \end{matrix} \right). NEWLINE\]NEWLINE Here \(Q\), \(R\) are operator polynomials, \(\varphi^n = (\varphi^n_1, \varphi^n_2)^T \in H\), \(\varphi_1^n = f_1^n\), and \(\varphi^n_2 = \frac{1}{p} Q^{-1}(P^{p-1}f_2^n + \cdots + Pf_2^{n + (p-2)/p} + f_2^{n + (p-1)/p})\), \(P = I_2 - \tau A_{22}\).NEWLINENEWLINENEWLINEThe following theorem holds:NEWLINENEWLINENEWLINETheorem. Let \(\Delta t\|A_{11}\|_{(1)}\leq 1 - \varepsilon\), \(\varepsilon\in (0,1)\), \(\tau\|A_{22}\|_{(2)}\leq 1\). Then in the case of zero initial data the following inequality holds: NEWLINE\[NEWLINE (Au^m,u^m)\leq \frac{1}{\varepsilon}\sum_{n=0}^{m-1}\Delta t\left( \|f_1^n\|^2 + 4\max_{k=1,\dots, p}\|f_2^{n + (k-1)/p}\|^2_2\right).NEWLINE\]
reviewed by: Vladimir N. Grebenev
description: scientific article; zbMATH DE number 1603285