  High order methods are often desired for the evolution of ordinary
differential equations, in particular those arising from the
semi-discretization of partial differential equations. In prior work in we
investigated the interplay between the local truncation error and the global
error to construct error inhibiting general linear methods (GLMs) that control
the accumulation of the local truncation error over time. Furthermore we
defined sufficient conditions that allow us to post-process the final solution
and obtain a solution that is two orders of accuracy higher than expected from
truncation error analysis alone. In this work we extend this theory to the
class of two-derivative GLMs. We define sufficient conditions that control the
growth of the error so that the solution is one order higher than expected from
truncation error analysis, and furthermore define the construction of a simple
post-processor that will extract an additional order of accuracy. Using these
conditions as constraints, we develop an optimization code that enables us to
find explicit two-derivative methods up to eighth order that have favorable
stability regions, explicit strong stability preserving methods up to seventh
order, and A-stable implicit methods up to fifth order. We numerically verify
the order of convergence of a selection of these methods, and the total
variation diminishing performance of some of the SSP methods. We confirm that
the methods found perform as predicted by the theory developed herein.
