  The long-standing challenge of creating a Milky Way-like disk galaxy from
cosmological simulations has motivated significant developments in both
numerical methods and physical models in recent years. We investigate these two
fundamental aspects in a new comparison project using a set of cosmological
hydrodynamic simulations of the formation and evolution of a Milky Way-size
galaxy. In this study, we focus on the comparison of two particle-based
hydrodynamics methods: the improved smoothed particle hydrodynamics (SPH) code
Gadget, and the Lagrangian Meshless Finite-Mass (MFM) code GIZMO. All the
simulations in this paper use the same initial conditions and physical models,
which include physics of both dark matter and baryons, star formation,
"energy-driven" outflow, metal-dependent cooling, stellar evolution and metal
enrichment from supernovae. We find that both numerical schemes produce a
late-type galaxy with extended gaseous and stellar disks. However, notable
differences are present in a wide range of galaxy properties and their
evolution, including star formation history, gas content, disk structure and
kinematics. In particular, there is significant difference in gas properties
and their evolution between the two simulations. Compared to GIZMO, Gadget
simulation produces a larger fraction of cold, dense gas at high redshift which
fuels rapid star formation and results in a higher stellar mass by $20\%$ and a
lower gas fraction by $10\%$ at $z = 0$, and the resulting gas disk is smoother
and more coherent in rotation due to damping of turbulent motion by the
numerical viscosity in SPH, in contrast to the GIZMO simulation which shows
more prominent spiral structure. Given its better convergence properties and
lower computational cost, we argue that MFM method is a promising alternative
to the widely used SPH in cosmological hydrodynamic simulations.
