A Nonintrusive Reduced-Order Model for Uncertainty Quantification in Numerical Solution of One-Dimensional Free-Surface Water Flows Over Stochastic Beds

Abstract

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.