  The transient start-up flow solution with slip is a useful tool to verify
Computational Fluid Dynamics (CFD) simulations. However, a highly accurate,
open-source black box solution does not seem to be available. Our method
provides a fast, automated, and rigorously verified open-source implementation
that can compute the hydrodynamic eigenmodes of a two-dimensional channel flow
beyond the standard floating-point precision. This allows for a very accurate
computation of the corresponding Fourier series solution. We prove that all
roots are found in all special cases for the general flow problem with
different slip lengths on the channel walls. The numerical results confirm
analytically derived asymptotic power laws for the leading hydrodynamic
eigenmode and the characteristic timescale in the limiting cases of small and
large slip. The code repository including test cases is publicly available here
https://git.rwth-aachen.de/fricke/start-up-flow
