  Implicit solvers present strong limitations when used on supercomputing
facilities and in particular for adaptive mesh-refinement codes. We present a
new method for implicit adaptive time-stepping on adaptive mesh
refinement-grids. We implement it in the radiation hydrodynamics solver we
designed for the RAMSES code for astrophysical purposes and, more particularly,
for protostellar collapse. We briefly recall the radiation hydrodynamics
equations and the adaptive time-stepping methodology used for hydrodynamical
solvers. We then introduce the different types of boundary conditions
(Dirichlet, Neumann, and Robin) that are used at the interface between levels
and present our implementation of the new method in the RAMSES code. The method
is tested against classical diffusion and radiation hydrodynamics tests, after
which we present an application for protostellar collapse. We show that using
Dirichlet boundary conditions at level interfaces is a good compromise between
robustness and accuracy and that it can be used in structure formation
calculations. The gain in computational time over our former unique time step
method ranges from factors of 5 to 50 depending on the level of adaptive
time-stepping and on the problem. We successfully compare the old and new
methods for protostellar collapse calculations that involve highly nonlinear
physics. We have developed a simple but robust method for adaptive
time-stepping of implicit scheme on adaptive mesh-refinement grids. It can be
applied to a wide variety of physical problems that involve diffusion
processes.
