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In this paper we generalize the polynomial time integration framework toadditively partitioned initial value problems. The framework we present isgeneral and enables the construction of many new families of additiveintegrators with arbitrary order-of-accuracy and varying degree ofimplicitness. In this first work, we focus on a new class of implicit-explicitpolynomial block methods that are based on fully-implicit Runge-Kutta methodswith Radau nodes, and possess high stage order. We show that the newfully-implicit-explicit (FIMEX) integrators have improved stability compared toexisting IMEX Runge-Kutta methods, while also being more computationallyefficient due to recent developments in preconditioning techniques for solvingthe associated systems of nonlinear equations. For PDEs on periodic domainswhere the implicit component is trivial to invert, we will show howparallelization of the right-hand-side evaluations can be exploited to obtainsignificant speedup compared to existing serial IMEX Runge-Kutta methods. Forparallel (in space) finite-element discretizations, the new methods can achieveorders of magnitude better accuracy than existing IMEX Runge-Kutta methods,and/or achieve a given accuracy several times times faster in terms ofcomputational runtime.

Additive Polynomial Time Integrators, Part I: Framework and Fully-Implicit-Explicit (FIMEX) Collocation Methods