Radiation hydrodynamics with Adaptive Mesh Refinement and application to prestellar core collapse. I Methods

Radiative transfer has a strong impact on the collapse and the fragmentation of prestellar dense cores. We present the radiation-hydrodynamics solver we designed for the RAMSES code. The method is designed for astrophysical purposes, and in particular for protostellar collapse. 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 photon). 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. We report in details our methodology to integrate the RHD solver into RAMSES. We test successfully the method against 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 compare successfully the results with previous studies using different models for radiation and hydrodynamics. 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.

💡 Research Summary

This paper presents a new radiation‑hydrodynamics (RHD) module for the adaptive‑mesh‑refinement (AMR) code RAMSES, specifically designed for simulations of low‑mass star formation. The authors adopt the grey (frequency‑integrated) flux‑limited diffusion (FLD) approximation in the comoving frame, using Miner‑bo’s flux limiter to smoothly transition between optically thick and thin regimes. The governing equations include the standard hydrodynamic conservation laws, augmented by a radiation pressure term (Pr = Er/3) and source terms describing energy exchange between matter and radiation (Planck and Rosseland opacities).

Numerically, the scheme is split in time. The hyperbolic part (mass, momentum, total energy) is integrated explicitly with RAMSES’s second‑order MUSCL Godunov solver, modified to incorporate the radiation pressure contribution. The characteristic wave speeds are consequently increased to √(γ P/ρ + 4 Er/9ρ), and the Courant‑Friedrichs‑Lewy (CFL) condition is adjusted accordingly. The diffusive part (∇·(c λ/κR ∇Er)) and the stiff matter‑radiation coupling term (κP ρ c (a T⁴ – Er)) are treated implicitly. To keep the implicit system linear, the authors linearize the T⁴ term by a first‑order Taylor expansion, assuming temperature changes are small over a single timestep. This yields a set of linear equations coupling the updated gas temperature and radiation energy density in each cell. The global linear system A x = b is solved with a conjugate‑gradient algorithm, preserving the second‑order spatial accuracy of the underlying finite‑volume method.

The implementation respects the AMR hierarchy: each refinement level uses the same implicit solver, and the flux limiter and opacities are evaluated using the temperature from the previous explicit step to maintain linearity. The authors validate the method with a suite of standard tests, including radiative shock tubes, radiation diffusion, and radiation‑hydrodynamics coupling problems, all of which reproduce reference solutions.

As a scientific demonstration, the authors simulate the three‑dimensional collapse of an isolated, non‑rotating 1 M⊙ prestellar core. The initial cloud is isothermal and spherical; AMR refinement reaches sub‑AU scales near the center. The results show the expected rise in central temperature due to radiative heating, and the formation of the first hydrostatic core with mass (~0.02 M⊙) and radius (~5 AU) consistent with earlier studies (e.g., Krumholz et al. 2007; Commerçon et al. 2010). The inclusion of radiation pressure in the explicit step leads to slightly faster propagation of radiative waves compared with purely diffusive treatments, confirming the importance of the coupled explicit‑implicit approach.

In conclusion, the paper delivers a robust, second‑order accurate RHD solver that integrates seamlessly with RAMSES’s AMR infrastructure, enabling high‑resolution studies of star formation where radiative feedback plays a crucial role. The authors outline future extensions to magnetohydrodynamics, rotating cores, and multi‑group (frequency‑dependent) radiation transport, which will further broaden the applicability of the method.