  A high order one-step ADER-WENO finite volume scheme with Adaptive Mesh
Refinement (AMR) in multiple space dimensions is presented. A high order
one-step time discretization is achieved using a local space-time discontinuous
Galerkin predictor method, while a high order spatial accuracy is obtained
through a WENO reconstruction. Thanks to the one-step nature of the underlying
scheme, the resulting algorithm can be efficiently imported within an AMR
framework on space-time adaptive meshes. We provide convincing evidence that
the presented high order AMR scheme behaves better than traditional second
order AMR methods. Tests are shown of the new scheme for nonlinear systems of
hyperbolic conservation laws, including the classical Euler equations and the
equations of ideal magnetohydrodynamics. The proposed scheme is likely to
become a useful tool in several astrophysical scenarios.
