Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations, 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 two-dimensional Riemann problem, Lax-Liu #6, and a three-dimensional 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.385S173S193Ecole Centrale de Marseille (ECM)Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP)Brazilian Research Council (CNPq), BrazilANR project "SiCoMHD"Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)