  We present a class of high order finite volume schemes for the solution of
non-conservative hyperbolic systems that combines the one-step ADER-WENO finite
volume approach with space-time adaptive mesh refinement (AMR). The resulting
algorithm, which is particularly well suited for the treatment of material
interfaces in compressible multi-phase flows, is based on: (i) high order of
accuracy in space obtained through WENO reconstruction, (ii) a high order
one-step time discretization via a local space-time discontinuous Galerkin
predictor method, and (iii) the use of a path conservative scheme for handling
the non-conservative terms of the equations. The AMR property with time
accurate local time stepping, which has been treated according to a
'cell-by-cell' strategy, strongly relies on the high order one-step time
discretization, which naturally allows a high order accurate and consistent
computation of the jump terms at interfaces between elements using different
time steps. The new scheme has been successfully validated on some test
problems for the Baer-Nunziato model of compressible multiphase flows.
