On stability of two-layer explicit schemes (Q2714022)

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\[NEWLINE (Au^m,u^m)\leq \frac{1}{\varepsilon}\sum_{n=0}^{m-1}\Delta t\left( \|f_1^n\|^2 + 4\max_{k=1,\dots, p}\|f_2^{n + (k-1)/p}\|^2_2\right).NEWLINE\]