A well-balanced reconstruction with bounded velocities for the shallow water equations by convex combination

Finite volume schemes for hyperbolic balance laws require a piecewise polynomial reconstruction of the cell averaged values, and a reconstruction is termed `well-balanced' if it is able to simulate steady states at higher order than time evolving states. For the shallow water system this involves reconstructing in surface elevation, to which modifications must be made as the fluid depth becomes small to ensure positivity, and for many reconstruction schemes a modification of the inertial field is also required to ensure the velocities are bounded. We propose here a reconstruction based on a convex combination of surface and depth reconstructions which ensures that the depth increases with the cell average depth. We also discuss how, for cells that are much shallower than their neighbours, reducing the variation in the reconstructed flux yields bounds on the velocities. This approach is generalisable to high order schemes, problems in multiple spacial dimensions, and to more complicated systems of equations. We present reconstructions and associated technical results for three systems, the standard shallow water equations, shallow water in a channel of varying width, and a shallow water model of a particle driven current. Positivity preserving time stepping is also discussed.