Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit-implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

翻译：非平稳问题柯西问题近似求解的稳定计算算法基于隐式时间逼近。通过问题算子的加性分解以及使用特定显式-隐式时间逼近的近似解组合，可降低耦合多维方程系统边值问题的计算成本。当前此技术仅适用于分解步骤较为简单的情形。本文提出一种通用方法，用于在有限维希尔伯特空间中构建演化方程的分解-组合算法。该方法基于两个核心步骤：在分解阶段对相应空间中的单位算子进行两种基本形式的分解，以及在组合阶段应用加性算子差分格式。通过构建空间变量标准分裂格式和区域加性格式（区域分解格式），以二阶抛物型方程边值问题为例阐述通用结论。