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a b s t r a c tThis work deals with the flow generated in a shock tube after the shock wave has reflected at the end wall. For a viscous fluid, a complex unsteady interaction takes place between the incident boundary layer and reflected shock wave. The numerical simulation of this complex flow requires both robust and accurate numerical schemes. In this work, we rely on the one-step high-order scheme recently proposed by Daru and Tenaud [Daru V, Tenaud C. High order one-step monotonicity preserving schemes for unsteady flow calculations. J Comput Phys 2004;193:563-94]. With this scheme, converged results are obtained for Reynolds numbers in the range 200-1000. The interaction mechanisms are carefully analyzed as well as the flow dynamics.

Abstract. We tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. It considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by stability constraints of the diffusive steps, but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far, out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are finally discussed. Key words.Reaction-diffusion equations, multi-scale reaction waves, operator splitting, adaptive multiresolution AMS subject classifications. 33K57, 35A18, 65M50, 65M081. Introduction. Numerical simulations of multi-scale phenomena are commonly used for modeling purposes in many applications such as combustion, chemical vapor deposition, or air pollution modeling. In general, all these models raise several difficulties created by the high number of unknowns, the wide range of temporal scales due to large and detailed chemical kinetic mechanisms, as well as steep spatial gradients associated with very localized fronts of high chemical activity. Furthermore, a natural stumbling block to perform 3D simulations with all scales resolution is either the unreasonably small time step due to stability requirements or the unreasonable memory requirements for implicit methods. In this context, one can consider various numerical strategies in order to treat the induced stiffness for time dependent * This research was supported by a fundamental project grant from ANR (French National Research Agency -ANR Blancs) Séchelles (project leader S. Descombes -2009Descombes - -2013, by a CNRS PEPS Maths-ST2I project MIPAC (project leader V. Louvet -2009Louvet - -2010, and by a DIGITEO RTRA project MUSE (project leader M. Massot -2010Massot - -2014 problems. The most natural id...