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Numerical simulations of magneto-convection have greatly expanded ourunderstanding of stellar interiors and stellar magnetism. Recently, fullycompressible hydrodynamical simulations of full-star models have demonstratedthe feasibility of studying the excitation and propagation of pressure andinternal gravity waves in stellar interiors, which would allow for a directcomparison with asteroseismological measurements. However, the impact ofmagnetic fields on such waves has not been taken into account yet inthree-dimensional simulations. We conduct a proof of concept for therealization of three-dimensional, fully compressible, magneto-hydrodynamicalnumerical simulations of stellar interiors with the RAMSES code. We adapted theRAMSES code to deal with highly subsonic turbulence, typical of stellarconvection, by implementing a well-balanced scheme in the numerical solver. Wethen ran and analyzed three-dimensional hydrodynamical andmagneto-hydrodynamical simulations with different resolutions of aplane-parallel convective envelope on a Cartesian grid. Both hydrodynamical andmagneto-hydrodynamical simulations develop a quasi-steady, turbulent convectionlayer from random density perturbations introduced over the initial profiles.The convective flows are characterized by small-amplitude fluctuations aroundthe hydrodynamical equilibrium of the stellar interior, which is preserved overthe whole simulation time. Using our compressible well-balanced scheme, we wereable to model flows with Mach numbers as low as $\mathcal{M} \sim 10^{-3}$, buteven lower Mach number flows are possible in principle. In themagneto-hydrodynamical runs, we observe an exponential growth of magneticenergy consistent with the action of a small-scale dynamo. (Abridged)

Toward fully compressible numerical simulations of stellar magneto-convection with the RAMSES code