A symmetric formulation and SUPG scheme for the shallow-water equations

Shallow-water flows with supercritical and subcritical subregions often exhibit numerical difficulties because of their associated hydraulic jumps (shock waves), steep layers, and fictitious oscillations. Analogous problems in gas dynamics have led to the recent development of a promising class of Petrov-Galerkin methods specifically designed for hyperbolic/incompletely parabolic systems, and are written in a symmetric conservation form. One of the major difficulties in the application of this class of methods to shallow water problems has been the unavailability of a suitable symmetric form of the governing equations. In the present work, this issue is addressed by introducing the total energy of the water column to motivate a change of variables which symmetrizes the shallow-water conservation system. Then, the one-dimensional case is considered and a time-accurate, streamline-upwind Petrov-Galerkin (SUPG) scheme is developed based on the proposed symmetric form. Numerical results illustrate the method and permit comparison with other schemes.