**5. Conclusions and perspectives**

In this book chapter, we have presented a multidimensional modelling for the numerical computation of ignition and propagation of combustion fronts during combustion synthesis of ceramic materials such as titanium-carbide. A detailed computational study was done in 1D slab/cylindrical/spherical geometry, 2D cylindrical/cartesian and 3D cartesian geometry to analyze the influence of the radiative boundary conditions over the induction time. The radiation contribution to the thermal conductivity was taken into account and the sensitivity of the induction time to several parameters such as the kinetics, wall temperature, phase change was carefully analyzed. Our numerical software Hephaïstos was presented. It implements an implicit finite-volume scheme for which error estimates, discrete maximum principles were reported and used to ensure the consistency of the numerical results. This modelling study was done for titanium carbide TiC. It can be applied to other ceramic materials such as silicium carbide-SiC-. Moreover in order to analyze the combustion front propagation for high Zeldovich number cases, an adaptive finite-volume scheme is required. A new monotonicity preserving refinement/derefinement conservative algorithm has been designed for multidimensional computations in various coordinate systems. This algorithm maintains the structured topology of the mesh. It is currently under implementation.
