We present the special use of domain decomposition, operator splitting and function decomposition combined with asymptotic analytical qualitative results to obtain, on parallel computers, efficient and accurate solvers [4] adapted to the nature of the solution of combustion fronts. We develop a technique to compute non periodics solutions with Fourier expansion. Our applications in gas combustion, combustion of solid as well as frontal polymerization [5, 6] are characterized by stiff fronts that propagate with nonlinear dynamics. The multiple scale phenomena under consideration lead to very intense computations [7]. Our motivation is to study the interaction between a mechanism of convective instability similar to Rayleigh Bénard instability and a mechanism of thermal instability well known in solid combustion [3, 9].