Radiative hydrodynamics simulations of red supergiant stars IV. Gray versus non-gray opacities

Context. Red supergiants are massive evolved stars that contribute extensively to the chemical enrichment of our Galaxy. It has been shown that convection in those stars produces large granules that cause surface inhomogeneities and shock waves in the photosphere. The understanding of their dynamics is crucial for unveiling the unknown mass-loss mechanism, their chemical composition, and their stellar parameters. Aims. We present a new generation of red supergiant simulations with a more sophisticated opacity treatment performed with 3D radiative-hydrodynamics code CO5BOLD. Methods. In the code the coupled equations of compressible hydrodynamics and non-local radiation transport are solved in the presence of a spherical potential. The stellar core is replaced by a special spherical inner boundary condition, where the gravitational potential is smoothed and the energy production by fusion is mimicked by a simply producing heat corresponding to the stellar luminosity. All outer boundaries are transmitting for matter and light. The post-processing radiative transfer code OPTIM3D is used to extract spectroscopic and interferometric observables. Results. We show that if one relaxes the assumption of frequency-independent opacities, this leads to a steeper mean thermal gradient in the optical thin region that strongly affects the atomic strengths and the spectral energy distribution. Moreover, the weaker temperature fluctuations reduce the incertitude on the radius determination with interferometry. We show that 1D models of red supergiants must include a turbulent velocity that is calibrated on 3D simulations to obtain the effective surface gravity that mimics the effect of turbulent pressure on the stellar atmosphere. We provide an empirical calibration of the ad hoc micro- and macroturbulence parameters for 1D models using the 3D simulations: we find that there is no clear distinction between the different macroturbulent profiles needed in 1D models to fit 3D synthetic