New techniques in designing finite-difference domain decomposition algorithm for the heat equation (Q1827256)

In recent ten years and more, parallel numerical methods for the heat equation have been studied. \textit{D. J. Evans} [Appl. Math. Modelling 9, 201--206 (1985; Zbl 0591.65068)] and \textit{B.-L. Zhang} [Chin. J. Numer. Math. Appl. 14, No. 3, 27--37 (1992; Zbl 0891.65098)] have developed a class of alternating schemes in three time levels which are the alternating group explicit and the alternating segment explicit implicit methods. Both methods are unconditionally stable and have the obvious property of parallelism, and the latter can be more accurate in practical computation. In the present paper, some new techniques have been developed by using smaller time step \(\Delta \overline {t}={\Delta t}/m\) (\(m\) is a positive integer) in Saul'yev schemes at the interface points. The algorithms designed with new techniques can increase the stability bounds of the classical explicit scheme by \(2m\) times, and their numerical solution satisfies the similar error estimates to obtained by \textit{C. N. Dawson}, \textit{Q. Du} and \textit{T. F. Dupont} [Math. Comput. 57, No. 195, 63--71 (1991; Zbl 0732.65091)]. The paper is organized as follows. In the Section 2, the authors construct some schemes, respectively, for \(m=2\) and \(m=3\) at the interface points. In the Section 3, the domain decomposition algorithms with the schemes in Section 2 are defined, for which the convergence results of the numerical solutions are obtained. In the Section 4, some numerical examples are given to show the stability and the accuracy of the algorithms.