  We perform global two-dimensional hydrodynamical simulations of Keplerian
discs with free eccentricity over thousands of orbital periods. Our aim is to
determine the validity of secular theory in describing the evolution of
eccentric discs, and to explore their nonlinear evolution for moderate
eccentricities. Linear secular theory is found to correctly predict the
structure and precession rates of discs with small eccentricities. However,
discs with larger eccentricities (and eccentricity gradients) are observed to
precess faster (retrograde relative to the orbital motion), at a rate that
depends on their eccentricities (and eccentricity gradients). We derive
analytically a nonlinear secular theory for eccentric gas discs, which explains
this result as a modification of the pressure forces whenever eccentric orbits
in a disc nearly intersect. This effect could be particularly important for
highly eccentric discs produced in tidal disruption events, or for narrow
gaseous rings; it might also play a role in causing some of the variability in
superhump binary systems. In two dimensions, the eccentricity of a moderately
eccentric disc is long-lived and persists throughout the duration of our
simulations. Eccentric modes are however weakly damped by their interaction
with non-axisymmetric spiral density waves (driven by the Papaloizou-Pringle
instability, which occurs in our idealised setup with solid walls), as well as
numerical diffusion.
