  Obtainable computational efficiency is evaluated when using an Adaptive Mesh
Refinement (AMR) strategy in time accurate simulations governed by sets of
conservation laws. For a variety of 1D, 2D, and 3D hydro- and
magnetohydrodynamic simulations, AMR is used in combination with several
shock-capturing, conservative discretization schemes. Solution accuracy and
execution times are compared with static grid simulations at the corresponding
high resolution and time spent on AMR overhead is reported. Our examples reach
corresponding efficiencies of 5 to 20 in multidimensional calculations and only
1.5 -- 8 % overhead is observed. For AMR calculations of multi-dimensional
magnetohydrodynamic problems, several strategies for controlling the $\nabla
\cdot \BB =0$ constraint are examined. Three source term approaches suitable
for cell-centered $\BB$ representations are shown to be effective. For 2D and
3D calculations where a transition to a more globally turbulent state takes
place, it is advocated to use an approximate Riemann solver based
discretization at the highest allowed level(s), in combination with the robust
Total Variation Diminishing Lax-Friedrichs method on the coarser levels. This
level-dependent use of the spatial discretization acts as a computationally
efficient, hybrid scheme.
