  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.
