  The present paper introduces a class of finite volume schemes of increasing
order of accuracy in space and time for hyperbolic systems that are in
conservation form. This paper specifically focuses on Euler system that is used
for modeling the flow of neutral fluids and the divergence-free, ideal
magnetohydrodynamics (MHD) system that is used for large scale modeling of
ionized plasmas.
  Efficient techniques for weighted essentially non-oscillatory (WENO)
interpolation have been developed for finite volume reconstruction on
structured meshes. We also present a new formulation of the ADER (for Arbitrary
Derivative Riemann Problem) schemes that relies on a local continuous
space-time Galerkin formulation instead of the usual Cauchy-Kovalewski
procedure.
  The schemes reported here have all been implemented in the RIEMANN framework
for computational astrophysics. We demonstrate that the ADER-WENO meet their
design accuracies. Several stringent test problems of Euler flows and MHD flows
are presented in one, two and three dimensions. Many of our test problems
involve near infinite shocks in multiple dimensions and the higher order
schemes are shown to perform very robustly and accurately under all conditions.
