Massively parallel computation of stiff propagating combustion fronts

Gas combustion, solid combustion as well as frontal polymerization are characterized by stiff fronts that propagate with nonlinear dynamics. The multiple-scale phenomena under consideration lead to very intense computations that require parallel computing in order to reduce the elapsed time of the computation. We develop a methodology to build on the MIMD architecture a parallel numerical method based on the property of the solution, i.e. a stiff quasi-planar two-dimensional combustion front. We illustrate our methodology using two models of the combustion process. The first is a thermo-diffusive model of a two-step chemical reaction exhibiting two transition layers. The second is a thermo-diffusive model of a one-step chemical reaction coupled with a hydrodynamical model using the stream function - vorticity formulation of the Navier - Stokes equations written in the Boussinesq approximation. This methodology makes use of efficient domain decomposition methods, combined with asymptotic analytical qualitative results to adapt the interface position, to solve the transition layer(s) of the solution accurately and operator splitting to take advantage of the quasi-planar property of the frontal process. Then, it provides three complementary levels of parallelism. A first level of parallelism based on the domain decomposition, thus a priori limited to the number of transition layers in the problem. A second based on an explicit parallelism in the orthogonal direction of the front propagation. A third based on the spread of equations on subnetworks of processors. The parallel implementation using the message passing library concept on the Paragon and iPSC860 MIMD computers are discussed. An efficient parallel algorithm to solve the space-periodic stream-function in the second model, based on Fourier modes decomposition combined with the first and second level of parallelism is provided. The direct numerical simulation provided by our numerical method allows us to explore the physical parameter space of the combustion process in order to understand the mechanism of instabilities. Some examples of hydrodynamical and thermal instabilities are given.