Strong horizontal gradients of density are responsible for the occurence of a large number of (often catastrophic) flows, such as katabatic winds, dust storms, pyroclastic flows and powder-snow avalanches. For a large number of applications, the overall density contrast in the flow remains small and simulations are carried in the Boussinesq limit, where density variations only appear in the body-force term. However, pyroclastic flows and powder-snow avalanches involve much larger density contrasts, which implies that the inhomogeneous Navier-Stokes equations need to be solved, along with a closure equation describing the mass diffusion. We propose a Lagrange-Galerkin numerical scheme to solve this system, and prove optimal error bounds subject to constraints on the order of the discretization and the time-stepping. Simulations of physical relevance are then shown.
