Coalescence of argon droplets with a radius of 25, 50, and 100 nm is studied with computational methods. Molecular dynamics (MD) simulations are carried out to generate reference data. Moreover, a phase-field model resting on a Helmholtz energy equation of state is devised and evaluated by computational fluid dynamics (CFD) simulations. Exactly the same scenarios in terms of geometry, fluid, and state are considered with these approaches. The MD and CFD simulation results show an excellent agreement over the entire coalescence process, including the decay of the inertia-induced oscillation of the merged droplet. Theoretical knowledge about the asymptotic behavior of coalescence process regimes is confirmed. All considered scenarios cross from the inertially limited viscous regime over to the inertial regime because of the low shear viscosity of argon. The particularly rapid dynamics during the initial stages of the coalescence process in the thermal regime is also captured by the phase-field model, where a closer look at the liquid density reveals that metastable states associated with negative pressure are attained in the emerging liquid bridge between the coalescing droplets. This demonstrates that this model is even capable of adequately handling the onset of coalescence. To speed up CFD simulations, the phase-field model is transferred to coarser grids through an interface widening approach that retains the thermodynamic properties including the surface tension.