POD-based model reduction for stabilized finite element approximations of shallow water flows

The shallow water equations (SWE) are used to model a wide range of free-surface flows from dam breaks and riverine hydrodynamics to hurricane storm surge and atmospheric processes. Despite their frequent use and improvements in algorithm and processor performance, accurate resolution of these flows is a computationally intensive task for many regimes. The resulting computational burden persists as a barrier to the inclusion of fully resolved two-dimensional shallow water models in many applications, particularly when the analysis involves optimal design, parameter inversion, risk assessment, and/or uncertainty quantification.Here, we consider model reduction for a stabilized finite element approximation of the SWE that can resolve advection-dominated problems with shocks but is also suitable for more smoothly varying riverine and estuarine flows. The model reduction is performed using Galerkin projection on a global basis provided by Proper Orthogonal Decomposition (POD). To achieve realistic speedup, we evaluate alternative techniques for the reduction of the non-polynomial nonlinearities that arise in the stabilized formulation. We evaluate the schemes' performance by considering their accuracy, robustness, and speed for idealized test problems representative of dam-break and riverine flows.