  This report presents a series of implicit-explicit (IMEX) variable timestep
algorithms for the incompressible Navier-Stokes equations (NSE). With the
advent of new computer architectures there has been growing demand for low
memory solvers of this type. The addition of time adaptivity improves the
accuracy and greatly enhances the efficiency of the algorithm. We prove energy
stability of an embedded first-second order IMEX pair. For the first order
member of the pair, we prove stability for variable stepsizes, and analyze
convergence. We believe this to be the first proof of this type for a variable
step IMEX scheme for the incompressible NSE. We then define and test a variable
stepsize, variable order IMEX scheme using these methods. Our work contributes
several firsts for IMEX NSE schemes, including an energy argument and error
analysis of a two-step, variable stepsize method, and embedded error estimation
for an IMEX multi-step method.
