A unified formulation of splitting-based implicit time integration schemes


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Academic Press – Elsevier


Splitting-based time integration approaches such as fractional step, 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 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.



Science & Technology, Technology, Physical Sciences, Computer Science, Interdisciplinary Applications, Physics, Mathematical, Computer Science, Physics, General-structure additive Runge&ndash, Kutta&nbsp, methods, Alternating direction implicit, Implicit-explicit, Implicit-implicit methods, RUNGE-KUTTA METHODS, APPROXIMATE MATRIX FACTORIZATION, FRACTIONAL STEP DISCRETIZATIONS, GENERAL LINEAR METHODS, W-METHODS, PARABOLIC PROBLEMS, ORDER CONDITIONS, PEER METHODS, STABILITY, EQUATIONS, 01 Mathematical Sciences, 02 Physical Sciences, 09 Engineering, Applied Mathematics