Efficient solvers for shallow-water Saint-Venant equations and debris transportation-deposition models

This research is aimed at achieving an efficient digital infrastructure for evaluating risks and damages caused by tsunami flooding. It is mainly focused on the suitable modeling of debris dynamics for a simple (but accurate enough) assessment of damages. For different reasons including computational performance and Big Data management issues, we focus our research on Eulerian debris flow modeling. Rather than using complex multiphase debris models, we rather use an empirical transportation and deposition model that takes into account the interaction with the main water flow, friction/contact with the ground but also debris interaction. In particular, for debris interaction, we have used ideas coming from vehicular traffic flow modeling. We introduce a velocity regularization term similar to the so-called ``anticipation term'' in traffic flow modeling that takes into account the local flow between neighboring debris and makes the problem mathematically well-posed. It prevents from the generation of ``Dirac measures of debris'' at shock waves. As a result, the model is able to capture emerging phenomenons like debris aggregation and accumulations, and possibly to react on the main flow by creating hills of debris and make the main stream deviate. We also discuss the way to derive quantities of interest (QoI), especially ``damage functions'' from the debris density and momentum fields. We believe that this original unexplored debris approach can lead to a valuable analysis of tsunami flooding damage assessment with Physics-based damage functions. Numerical experiments show the nice behaviour of the numerical solvers, including the solution of Saint-Venant's shallow water equations and debris dynamics equations.