High-order well-balanced schemes for shallow models for dry avalanches

In this work we consider a depth-averaged model for granular flows with a Coulomb-type friction force described by the $\mu \left(I\right)$ rheology. In this model, the so-called lake-at-rest steady states are of special interest, where velocity is zero and the slope is under a critical threshold defined by the angle of repose of the granular material. It leads to a family with an infinite number of lake-at-rest steady states. We describe a well-balanced reconstruction procedure that allows to define well-balanced finite volume methods for such problem. The technique is generalized to high-order space/time schemes. In particular, the second and third-order schemes are considered in the numerical tests section. An accuracy test is included showing that second and third-order are achieved. A well-balanced test is also considered. The proposed scheme is well-balanced for steady states with non-constant free surface, and it is exactly well-balanced for those steady states given by a simple characterization.