Adaptive numerical methods in time and space for the simulation of multi-scale reaction fronts. (Méthodes numériques adaptatives pour la simulation de la dynamique de fronts de réaction multi-échelles en temps et en espace)

We tackle the development of a new generation of numerical methods for the solution of time dependent PDEs modeling general time/space multi-scale phenomena issued from various application fields. This type of problem induces well-known numerical restrictions and potentially large stiffness, which stem from the broad spectrum of time scales in the nonlinear chemical terms as well as from steep, spatially very localized, spatial gradients in the reaction fronts. Therefore, dedicated numerical strategies are needed to ensure the accuracy of the numerical approximations from a theoretical point of view, taking also into account adequate practical implementations to reduce computational costs. In order to cope with these problems, this study introduces a few mathematical and numerical elements for the solution of stiff reaction-diffusion systems, extensible in practice to more general configurations. The core of the numerical strategy is thus based on a specially conceived operator splitting method with dedicated high order time integration schemes for each subproblem. An appropriate choice of splitting time steps allows us the simulation of the solution within a prescribed accuracy, according to the overall physics of the problem. The resulting numerical scheme is properly coupled with an adaptive multiresolution technique for dynamic spatial mesh representations of the solution. Such an approach has led to the conception of the academic, generic 1D/2D/3D MBARETE code in order to evaluate the proposed theoretical and numerical developments in practical stiff configurations arising in several research fields. The algorithmic efficiency of the method is assessed by the simulation of propagating stiff reaction waves issued from nonlinear chemical dynamics and from biomedical engineering applications for a brain stroke model with "detailed chemical mechanisms''. Moreover, in order to extend the applicability of the method to more complex and unsteady problems, we consider for the first time a time adaptive splitting scheme for stiff PDEs, that yields dynamic time stepping within the prescribed accuracy. The fully time/space adaptive method allows us then a consistent description of reaction-diffusion-convection problems disclosing a broad spectrum of time/space scales as well as different physical scenarios, such as highly nanosecond repetitively pulsed discharges or self-ignition and propagation of flames for, respectively, plasma and combustion applications. The main goal of this work is hence to numerically solve stiff PDEs with reasonable, standard computational resources and based on a mathematical background that ensures robust, general and accurate numerical schemes. Further studies are also presented that include time parallelization strategies, parallel computing techniques for shared memory architectures and complementary mathematical characterization of splitting schemes.