Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge--Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions and stability analysis for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

翻译：基于分解的时间一体化办法,如分步制、交替方向隐含、操作员分裂和地方一维方法,将利益系统分成各组成部分,并以具有成本效益的方式间接解决个别组成部分。这项工作提议在一般结构添加剂龙格-库塔(GARK)方法的框架内统一制定分时间一体化办法。具体地说,我们制定隐含(IMIM)的GARK办法,为这一类提供秩序条件和稳定性分析,并解释其对普通差分方程式分隔系统的应用。我们表明传统的分解方法属于IMGARK家族,因此可以在这个统一的框架内加以研究。新的IMIM-GARK的分解方法是利用抛物法系统开发和测试的。