Intrusive polynomial chaos uncertainty quantification on shallow water system

In order to build a surrogate model and study the propagation of uncertainty in hydrological systems, it usually takes significant time to determine a sampling strategy and run hundreds of forward simulations. The challenge comes when a timely prediction is required in the presence of uncertainties. One such situation is predicting storm surge during hurricanes. We develop a stochastic shallow water model to resolve this issue. A stochastic Galerkin method is used to do the probability space discretization. An incremental pressure correction scheme is used to decouple the shallow water equations. In order to resolve the issue in the resulting stochastic hyperbolic system, we propose generalizing each stabilization method into the probability space by adding stabilization terms to every stochastic mode in a modes-coupled way. We first verify the deterministic part of this model by comparing it against analytical solutions as well as solutions from other well-established shallow water models. Then, we verify the predicted probability density function given by the stochastic model through a two-way comparison in several different test cases. Finally, the model is used to study second-order stochastic processes. We demonstrate that the maximum variance occurs at the extreme mean for both surface elevation and water velocity over space and over time. The model is also used to predict the uncertainty under an uncertain wind drag coefficient during historical hurricane events. We conclude that the wind drag coefficient has a significant impact on the surface elevation, at least up to an offset of 0.66 meters, given an uncertain wind drag coefficient in two scenarios. Furthermore, we propose a predictor, μ + σ, and demonstrate its effectiveness under the same uncertain wind drag coefficient.Engineering Mechanic