The study of mechanisms that allow the formation of structures such as sand dunes and ripples at the bottom of a fluid flow plays a crucial role in the understanding of coastal dynamics. The modeling of these phenomena is particularly complex since we must not only solve the Navier-Stokes or Saint-Venant equations with equation for sediment transport, but also take into account the evolution of...

Extending results of Oh–Zumbrun and Johnson–Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, w...

Conditions for boundary feedback stabilizability of non-uniform linear 2 × 2 hyperbolic systems over a bounded interval are investigated. Themain result is to show that the existence of a basic quadratic control Lyapunov function requires that the solution of an associated ODE is defined on the considered interval. This result is used to give explicit conditions for the existence of stabilizing...

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves stationary steady states (lake at rest) and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these fe...