  We study the nuclear symmetry energy S(rho) and related quantities of nuclear
physics and nuclear astrophysics predicted generically by relativistic
mean-field (RMF) and Skyrme-Hartree-Fock (SHF) models. We establish a simple
prescription for preparing equivalent RMF and SHF parametrizations starting
from a minimal set of empirical constraints on symmetric nuclear matter,
nuclear binding energy and charge radii, enforcing equivalence of their Lorenz
effective masses, and then using the pure neutron matter (PNM) equation of
state (EoS) obtained from ab-initio calculations to optimize the pure isovector
parameters in the RMF and SHF models. We find the resulting RMF and SHF
parametrizations give broadly consistent predictions of the symmetry energy J
and its slope parameter L at saturation density within a tight range of <~2 MeV
and <~6 MeV respectively, but that clear model dependence shows up in the
predictions of higher-order symmetry energy parameters, leading to important
differences in (a) the slope of the correlation between J and L from the
confidence ellipse, (b) the isospin-dependent part of the incompressibility of
nuclear matter K_tau, (c) the symmetry energy at supra-saturation densities,
and (d) the predicted neutron star radii. The model dependence can lead to
about 1-2 km difference in predictions of the neutron star radius given
identical predicted values of J, L and symmetric nuclear matter (SNM)
saturation properties. Allowing the full freedom in the effective masses in
both models leads to constraints of 30<~J<~31.5 MeV, 35<~L<~60 MeV,
-330<~K_tau<~-216 MeV for the RMF model as a whole and 30<~J<~33 MeV, 28<~L<~65
MeV, -420<~K_tau<~-325 MeV for the SHF model as a whole. Notably, given PNM
constraints, these results place RMF and SHF models as a whole at odds with
some constraints on K_tau inferred from giant monopole resonance and neutron
skin experimental results.
