Radiation hydrodynamics with adaptive mesh refinement and application to prestellar core collapse. I. Methods

International audienceContext. Radiative transfer has a strong impact on the collapse and the fragmentation of prestellar dense cores. Aims: We present the radiation-hydrodynamics (RHD) solver we designed for the RAMSES code. The method is designed for astrophysical purposes, and in particular for protostellar collapse. Methods: We present the solver, using the co-moving frame to evaluate the radiative quantities. We use the popular flux-limited diffusion approximation under the grey approximation (one group of photons). The solver is based on the second-order Godunov scheme of RAMSES for its hyperbolic part and on an implicit scheme for the radiation diffusion and the coupling between radiation and matter. Results: We report in detail our methodology to integrate the RHD solver into RAMSES. We successfully test the method in several conventional tests. For validation in 3D, we perform calculations of the collapse of an isolated 1 M&sun; prestellar dense core without rotation. We successfully compare the results with previous studies that used different models for radiation and hydrodynamics. Conclusions: We have developed a full radiation-hydrodynamics solver in the RAMSES code that handles adaptive mesh refinement grids. The method is a combination of an explicit scheme and an implicit scheme accurate to the second-order in space. Our method is well suited for star-formation purposes. Results of multidimensional dense-core-collapse calculations with rotation are presented in a companion paper