  In simulations of fluid motion time accuracy has proven to be elusive. We
seek highly accurate methods with strong enough stability properties to deal
with the richness of scales of many flows. These methods must also be easy to
implement within current complex, possibly legacy codes. Herein we develop,
analyze and test new time stepping methods addressing these two issues with the
goal of accelerating the development of time accurate methods addressing the
needs of applications. The new methods are created by introducing inexpensive
pre-filtering and post-filtering steps to popular methods which have been
implemented and tested within existing codes. We show that pre-filtering and
post-filtering a multistep or multi-stage method results in new methods which
have both multiple steps and stages: these are general linear methods (GLMs).
We utilize the well studied properties of GLMs to understand the accuracy and
stability of filtered method, and to design optimal new filters for popular
time-stepping methods. We present several new embedded families of high
accuracy methods with low cognitive complexity and excellent stability
properties. Numerical tests of the methods are presented, including ones
finding failure points of some methods. Among the new methods presented is a
novel pair of alternating filters for the Implicit Euler method which induces a
third order, A-stable, error inhibiting scheme which is shown to be
particularly effective.
