ESAIM: PROCEEDINGS, December 2009, Vol. 29, p. 28-42 F. Coquel, Y. Maday, S. M¨ uller, M. Postel and Q. H. Tran, Editors ADAPTIVE MULTIRESOLUTION OR ADAPTIVE MESH REFINEMENT? A CASE STUDY FOR 2D EULER EQUATIONS Ralf Deiterding 1 , Margarete O. Domingues 2, 3 , Sˆ onia M. Gomes 4 , Olivier Roussel 3, 4 and Kai Schneider 53, 5 R´ esum´ e. Nous pr´ esentons des simulations adaptatives multir´ esolution (MR) des ´ equations d’Euler compressibles bi-dimensionnelles pour un probl` eme de Riemann classique. Les r´ esultats sont compar´ es en pr´ ecision et en efficacit´ e – temps CPU et place m´ emoire – avec ceux obtenus par la m´ ethode volumes finis sur la grille la plus fine. Pour le mˆ eme cas-test, nous pr´ esentons les calculs obtenus `a l’aide de la m´ ethode AMR (Adaptive Mesh Refinement) en imposant les mˆ emes crit` eres de pr´ ecision. Les r´ esultats ainsi obtenus sont compar´ es en termes d’effort de calcul et de compression m´ emoire, en utilisant des pas de temps globaux puis locaux. De ces r´ esultats pr´ eliminaires, nous concluons que les techniques multir´ esolution pr´ esentent des gains en termes de temps CPU et de place m´ emoire sup´ erieurs `a ceux de la m´ ethode AMR. Abstract. We present adaptive multiresolution (MR) computations of the two-dimensional com- pressible Euler equations for a classical Riemann problem. The results are then compared with respect to accuracy and computational efficiency, in terms of CPU time and memory requirements, with the corresponding finite volume scheme on a regular grid. For the same test case, we also perform compu- tations using adaptive mesh refinement (AMR) imposing similar accuracy requirements. The results thus obtained are compared in terms of computational overhead and compression of the computational grid, using in addition either local or global time stepping strategies. We preliminarily conclude that the multiresolution techniques yield improved memory compression and gain in CPU time with respect to the adaptive mesh refinement method. Introduction Adaptive discretization methods for solving nonlinear PDEs have a long tradition and can be tracked back to the late seventies [11]. Adaptivity is motivated by the huge computational complexity of real world problems which involve typically a multitude of dynamically active spatial and temporal scales. Introducing adaptivity 1 Computer Science and Mathematics Division, Oak Ridge National Laboratory, P.O. Box 2008 MS-6367, Oak Ridge, TN 37831, United States (e-mail : deiterdingr@ornl.gov) 2 Laborat´orio Associado de Computa¸ c˜ao e Matem´ atica Aplicada (LAC), Coordenadoria dos Laborat´ orios Associados (CTE), Instituto Nacional de Pesquisas Espaciais (INPE), Av. dos Astronautas 1758, 12227-010 S˜ ao Jos´ e dos Campos, S˜ ao Paulo, Brazil (e-mail : margarete@lac.inpe.br) 3 Laboratoire de Mod´ elisation en M´ ecanique et Proc´ ed´ es Propres (M2P2), CNRS, Universit´ es d’Aix-Marseille et Ecole Centrale Marseille, 38 rue F. Joliot-Curie, 13451 Marseille Cedex 20, France (e-mail : o roussel@yahoo.fr) 4 Universidade Estadual de Campinas (UNICAMP), IMECC, Caixa Postal 6065, 13083-970 Campinas, S˜ ao Paulo, Brazil (e- mail : soniag@ime.unicamp.br) 5 Centre de Math´ ematiques et d’Informatique (CMI), Universit´ e de Provence, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France (e-mail : kschneid@cmi.univ-mrs.fr) c  EDP Sciences, SMAI 2009 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc/2009053