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The long-standing challenge of creating a Milky Way-like disk galaxy fromcosmological simulations has motivated significant developments in bothnumerical methods and physical models in recent years. We investigate these twofundamental aspects in a new comparison project using a set of cosmologicalhydrodynamic simulations of the formation and evolution of a Milky Way-sizegalaxy. In this study, we focus on the comparison of two particle-basedhydrodynamics methods: the improved smoothed particle hydrodynamics (SPH) codeGadget, and the Lagrangian Meshless Finite-Mass (MFM) code GIZMO. All thesimulations 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 metalenrichment from supernovae. We find that both numerical schemes produce alate-type galaxy with extended gaseous and stellar disks. However, notabledifferences are present in a wide range of galaxy properties and theirevolution, including star formation history, gas content, disk structure andkinematics. In particular, there is significant difference in gas propertiesand their evolution between the two simulations. Compared to GIZMO, Gadgetsimulation produces a larger fraction of cold, dense gas at high redshift whichfuels rapid star formation and results in a higher stellar mass by $20\%$ and alower gas fraction by $10\%$ at $z = 0$, and the resulting gas disk is smootherand more coherent in rotation due to damping of turbulent motion by thenumerical viscosity in SPH, in contrast to the GIZMO simulation which showsmore prominent spiral structure. Given its better convergence properties andlower computational cost, we argue that MFM method is a promising alternativeto the widely used SPH in cosmological hydrodynamic simulations.

The Formation of a Milky Way-sized Disk Galaxy 1. A Comparison of Numerical Methods