Projection-Based Model Reduction for Finite Element Approximation of Shallow Water Flows

Abstract

Abstract. The shallow water equations (SWE) are used to model a wide range of environmental flows from dam breaks and riverine hydrodynamics to hurricane storm surge and atmospheric processes. Despite significant gains in numerical model efficiency stemming from algorithmic and hardware improvements, accurate shallow water modeling can still be very computationally intensive. The resulting computational expense remains 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 projection-based model reduction as a way to alleviate the computational burden associated with high-fidelity shallow-water approximations in ensemble forecast and sampling methodologies. In order to develop a robust approach that can resolve advectiondominated problems with shocks as well as more smoothly varying riverine and estuarine flows, we consider techniques using both Galerkin and Petrov-Galerkin projection on global bases provided by Proper Orthogonal Decomposition (POD). To achieve realistic speedup, we consider alternative methods for the reduction of the non-polynomial nonlinearities that arise in typical finite element formulations. We evaluate the schemes' performance by considering their accuracy and robustness for test problems in one and two space dimensions.