  We present a detailed comparison between two adaptive numerical approaches to
solve partial differential equations (PDEs), adaptive multiresolution (MR) and
adaptive mesh refinement (AMR). Both discretizations are based on finite
volumes in space with second order shock-capturing, and explicit time
integration either with or without local time-stepping. The two methods are
benchmarked for the compressible Euler equations in Cartesian geometry. As test
cases a 2D Riemann problem, Lax-Liu 6, and a 3D ellipsoidally expanding shock
wave have been chosen. We compare and assess their computational efficiency in
terms of CPU time and memory requirements. We evaluate the accuracy by
comparing the results of the adaptive computations with those obtained with the
corresponding FV scheme using a regular fine mesh. We find that both approaches
yield similar trends for CPU time compression for increasing number of
refinement levels. MR exhibits more efficient memory compression than AMR and
shows slightly enhanced convergence; however, a larger absolute overhead is
measured for the tested codes.
