  In this paper, we present a new method to perform numerical simulations of
astrophysical MHD flows using the Adaptive Mesh Refinement framework and
Constrained Transport. The algorithm is based on a previous work in which the
MUSCL--Hancock scheme was used to evolve the induction equation. In this paper,
we detail the extension of this scheme to the full MHD equations and discuss
its properties. Through a series of test problems, we illustrate the
performances of this new code using two different MHD Riemann solvers
(Lax-Friedrich and Roe) and the need of the Adaptive Mesh Refinement
capabilities in some cases. Finally, we show its versatility by applying it to
two completely different astrophysical situations well studied in the past
years: the growth of the magnetorotational instability in the shearing box and
the collapse of magnetized cloud cores. We have implemented this new Godunov
scheme to solve the ideal MHD equations in the AMR code RAMSES. It results in a
powerful tool that can be applied to a great variety of astrophysical problems,
ranging from galaxies formation in the early universe to high resolution
studies of molecular cloud collapse in our galaxy.
