Resumen de la Tesis:

This work presents a study of avalanche models using well balanced finite volumes schemes. For the physical model, we use the Saint-Venant system in two dimensions, including the effect of the friction as a source term. The model for the friction is the Voellmy-Salm rheological model. The numerical model was implemented in parts: in one dimension without friction, two dimensions without friction, and two dimension with friction. First we developed this scheme in one dimension. We use the hydrostatic reconstruction scheme. In this scheme, we reconstruct the states of the solution to compute the numerical fluxes, using a modified form of the stationary states relations. This scheme is well balanced, consistent and stable. For the two-dimensional problem, we use that the system is invariant under rotation. In this way, the numerical fluxes can be computed with a one-dimensional problem in which we apply the hydrostatic reconstruction scheme. This scheme is well balanced and consistent. To add the effect of the friction, we treat each part differently. The Coulomb friction can be included in the numerical scheme by modifying the source term in the hydrostatic reconstruction scheme. The turbulent friction must be included by a splitting method to ensure stability. We tested this scheme with the Rigopiano avalanche, a natural disaster that has been studied extensively. The real avalanches present steep terrain, so we have to use a global coordinate system to obtain better numerical results. The two-dimensional model with Voellmy-Salm friction in global coordinate system was presented. We tested the scheme with this new model for the Rigopiano avalanche. We also study ways to improve the numerical method. We show the Osher-Solomon method for the one-dimensional model in global coordinates. We show the hydrostatic scheme with the HLLC flux in one dimension and in two dimensions. We show a second order extension for the one-dimensional problem.