  The transient Couette and Poiseuille channel flow solutions with different
Navier slip conditions on the wall boundaries require the solution of a
non-linear root-finding problem. A novel algorithm is developed that automates
the computation of these roots with arbitrary precision and for general input
parameters. The obtained results show a significant improvement to available
coefficient values for this analytic solution. A parameter variation for the
slip lengths reveals a new power law for the first series coefficient that
governs the stability of the solution and is essential for slip length
computations from molecular dynamics simulations. Based on the new algorithm,
the time scales for the asymptotic approach to the stationary solution are
quantified. A double-precision implementation and an arbitrary precision
implementation provide a fast, verified, and highly accurate method to obtain
benchmark quality reference solutions or a module for more general
applications.
