  In interaction-dominated two-dimensional electron gases at intermediate
temperatures, electron transport is not diffusive as in the conventional Drude
picture but instead hydrodynamic. The relevant transport coefficient in this
regime is the shear viscosity. Here, we develop a numerically exact basis
expansion to solve the Fermi liquid equation, and apply it to compute the shear
viscosity of the electron gas with screened Coulomb interactions. Our
calculations are valid at all temperatures and in particular describe the
response beyond the asymptotic low-temperature limit, where perturbative
approaches exist. We show that even in this low-temperature limit, there is a
nonanalytic exchange contribution to the shear viscosity, highlighting the need
for a full nonperturbative solution of the Fermi liquid equation. We hope that
the techniques developed in this work will serve as a platform to determine the
response of interacting Fermi liquids.
