  In this work, we construct novel discretizations for the unsteady
convection-diffusion equation. Our discretization relies on multiderivative
time integrators together with a novel discretization that reduces the total
number of unknowns for the solver. These type of temporal discretizations come
from an umbrella class of methods that include Lax-Wendroff (Taylor) as well as
Runge-Kutta methods as special cases. We include two-point collocation methods
with multiple time derivatives as well as a sixth-order fully implicit
collocation method that only requires a total of three stages. Numerical
results for a number of sample linear problems indicate the expected order of
accuracy and indicate we can take arbitrarily large time steps.

Numerical Analysis