  This work generalizes 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 family 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
family, named GARK, introduces additional flexibility when compared to
traditional partitioned Runge-Kutta methods, and therefore offers additional
opportunities for the development of flexible solvers for systems with multiple
scales, or driven by multiple physical processes.
