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...

To cite this version:Max Duarte, Stéphane Descombes, Christian Tenaud, Sébastien Candel, Marc Massot. Time-space adaptive numerical methods for the simulation of combustion fronts. Combustion and Flame, Elsevier, 2013, 160 (6) AbstractThis paper presents a new computational strategy for the simulation of combustion fronts based on adaptive time operator splitting and spatial multiresolution. High-order and dedicated onestep solvers compose the splitting scheme for the reaction, diffusion, and convection subproblems, to independently cope with their inherent numerical difficulties and to properly solve the corresponding temporal scales. Adaptive and thus highly compressed spatial representations for localized fronts originating from multiresolution analysis result in important reductions of memory usage, and hence numerical simulations with sufficiently fine spatial resolution can be performed with standard computational resources. The computational efficiency is further enhanced by splitting time steps established beyond standard stability constraints associated to mesh size or stiff source time scales. The splitting time steps are chosen according to a dynamic splitting technique relying on solid mathematical foundations, which ensures error control of the time integration and successfully discriminates time-varying multi-scale physics. For a given semidiscretized problem, the solution scheme provides dynamic accuracy estimates that reflect the quality of numerical results in terms of numerical errors of integration and compressed spatial representations, for general multi-dimensional problems modeled by stiff PDEs. The strategy is efficiently applied to simulate the propagation of laminar premixed flames interacting with vortex structures, as well as various configurations of self-ignition processes of diffusion flames in similar vortical hydrodynamics fields. A detailed study of the error control is provided and show the potential of the approach. It yields large gains in CPU time, while consistently describing a broad spectrum of space and time scales as well as different physical scenarios.

In silico research in medicine is thought to reduce the need for expensive clinical trials under the condition of reliable mathematical models and accurate and efficient numerical methods. In the present work, we tackle the numerical simulation of reaction-diffusion equations modeling human ischemic stroke. This problem induces peculiar difficulties like potentially large stiffness which stems 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. Furthermore, simulations on realistic 3D geometries are mandatory in order to describe correctly this type of phenomenon. The main goal of this article is to obtain, for the first time, 3D simulations on realistic geometries and to show that the simulation results are consistent with those obtain in experimental studies or observed on MRI images in stroke patients.For this purpose, we introduce a new resolution strategy based mainly on time operator splitting that takes into account complex geometry coupled with a well-conceived parallelization strategy for shared memory architectures. We consider then a high order implicit time integration for 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. Thus, we aim at solving complete and realistic models including all time and space scales with conventional computing resources, that is on a reasonably powerful workstation. Consequently and as expected, 2D and also fully 3D numerical simulations of ischemic strokes for a realistic brain geometry, are conducted for the first time and shown to reproduce the dynamics observed on MRI images in stroke patients. Beyond this major step, in order to improve accuracy and computational efficiency of the simulations, we indicate how the present numerical strategy can be coupled with spatial adaptive multiresolution schemes. Preliminary results in the framework of simple geometries allow to assess the proposed strategy for further developments.