A unified formulation of splitting-based implicit time integration schemes

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 solve individual implicitly in a cost-effective way. This work proposes unified formulation splitting schemes framework general-structure additive Runge-Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide order conditions stability analysis for this class, explain their application to partitioned systems ordinary differential equations. We show that classical belong IMIM family, therefore can be studied framework. New IMIM-GARK are developed tested using parabolic systems.