A High-Order Finite Volume Method for Nonconservative Problems and Its Application to Model Submarine Avalanches

In this chapter we investigate how to apply a high-order finite volume method to discretize the model proposed in [FeBo08] to study submarine avalanches. The model proposed by Fernández-Nieto et al. in [FeBo08] is an integrated two-layer model of Savage–Hutter type. The upper layer models the fluid, and the second layer is assumed to be constituted by sediment or rocks. The derivation of the model is done by taking into account some physical properties of both layers: density, porosity, friction angle in a Coulomb law, internal friction angle between particles, and buoyancy. The previous model reduces to the one proposed by Savage and Hutter to study avalanches of granular materials when the height of the water layer tends to zero. In the pioneering works of Savage and Hutter (see [SaHu91]) a model to study avalanches over an inclined slope is proposed. They derive their model by integration of Euler equations and assuming a Coulomb friction law. Bouchut et al. in [BoMa03] propose a generalization of the model in order to take into account more general topographies. In particular, the angle of the bottom with the horizontal is not constant and depends on spatial variables. They show that a new term depending on the curvature is necessary to be introduced in the model in order to preserve stationary solutions and to verify an entropy inequality.