  High order implicit-explicit (IMEX) methods are often desired when evolving
the solution of an ordinary differential equation that has a stiff part that is
linear and a non-stiff part that is nonlinear. This situation often arises in
semi-discretization of partial differential equations and many such IMEX
schemes have been considered in the literature. The methods considered usually
have a a global error that is of the same order as the local truncation error.
More recently, methods with global errors that are one order higher than
predicted by the local truncation error have been devised (by Kulikov and
Weiner, Ditkowski and Gottlieb). In prior work we investigated the interplay
between the local truncation error and the global error to construct explicit
and implicit {\em error inhibiting schemes} that control the accumulation of
the local truncation error over time, resulting in a global error that is one
order higher than expected from the local truncation error, and which can be
post-processed to obtain a solution which is two orders higher than expected.
In this work we extend our error inhibiting with post-processing framework
introduced in our previous work to a class of additive general linear methods
with multiple steps and stages. We provide sufficient conditions under which
these methods with local truncation error of order p will produce solutions of
order (p+1), which can be post-processed to order (p+2), and describe the
construction of one such post-processor. We apply this approach to obtain
implicit-explicit (IMEX) methods with multiple steps and stages. We present
some of our new IMEX methods and show their linear stability properties, and
investigate how these methods perform in practice on some numerical test cases.
