  In this paper we generalize the polynomial time integration framework to
additively partitioned initial value problems. The framework we present is
general and enables the construction of many new families of additive
integrators with arbitrary order-of-accuracy and varying degree of
implicitness. In this first work, we focus on a new class of implicit-explicit
polynomial block methods that are based on fully-implicit Runge-Kutta methods
with Radau nodes, and possess high stage order. We show that the new
fully-implicit-explicit (FIMEX) integrators have improved stability compared to
existing IMEX Runge-Kutta methods, while also being more computationally
efficient due to recent developments in preconditioning techniques for solving
the associated systems of nonlinear equations. For PDEs on periodic domains
where the implicit component is trivial to invert, we will show how
parallelization of the right-hand-side evaluations can be exploited to obtain
significant speedup compared to existing serial IMEX Runge-Kutta methods. For
parallel (in space) finite-element discretizations, the new methods can achieve
orders of magnitude better accuracy than existing IMEX Runge-Kutta methods,
and/or achieve a given accuracy several times times faster in terms of
computational runtime.
