A Generalized-Structure Approach to Additive Runge-Kutta Methods

This work considers a general structure of the additively partitioned Runge--Kutta methods by allowing for different stage values as arguments of different components of the right-hand side. An order conditions theory is developed for the new formulation of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit and implicit-implicit methods in the new framework. The new approach, named GARK, introduces additional flexibility when compared to traditional partitioned Runge--Kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.

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