The D'yakonov fully explicit variant of the iterative decomposition method (Q1612516)

The paper is devoted to numerical methods for the simplest heat equation \(\frac{\partial u}{\partial t }=\frac{\partial^2 u}{\partial x^2 }\). Numerical examples are provided on a very coarse grid with \(h=1/10\) for several difference methods and several known smooth solutions. One of the implicit methods has accuracy \(O(h^4+[\Delta t]^2)\) and was known at least 70 years ago (including the two-dimensional case). The arising one-dimensional difference systems are (and have been for a long time) of no serious problem. It seems that the authors want to apply inner iterations of a structure that were used by me for more involved systems and can be of help for parallel computations. The authors make use of a simple factorization procedure with explicit formulas in the algorithm --- by this reason they write about the ``fully explicit'' method in contrast with the obvious implicit nature of the scheme. It is not clear why the authors prefer iterations with fixed parameter. The reference [[1]: \textit{E. G. D'yakonov}, Zh. Vychisl. Mat. Mat. Fiz. 3, 385-388 (1963; Zbl 0208.42301)] is not quite correct since in [1] the emphasis was on nonrectangular regions and general iterations. The corresponding factorization procedure (split difference operators), schemes with \(O(h^4+[\Delta t]^2)\) accuracy and also inner iterations were considered in other papers --- the references can be found in the book of \textit{E. G. D'yakonov} [Optimization in solving elliptic problems. (1996; Zbl 0852.65087)].