Stochastic Galerkin finite volume shallow flow model: well-balanced treatment over uncertain topography

Stochastic Galerkin methods can quantify uncertainty at a fraction of the computational expense of conventional Monte Carlo techniques, but such methods have rarely been studied for modeling shallow water flows. Existing stochastic shallow flow models are not well-balanced, and their assessment has been limited to stochastic flows with smooth probability distributions. This paper addresses these limitations by formulating a one-dimensional stochastic Galerkin shallow flow model using a low-order Wiener-Hermite polynomial chaos expansion with a finite volume Godunov-type approach, incorporating the surface gradient method to guarantee well-balancing. Preservation of a lake at rest over uncertain topography is verified analytically and numerically. The model is also assessed using flows with discontinuous and highly non-Gaussian probability distributions. Prescribing constant inflow over uncertain topography, the model converges on a steady-state flow that is subcritical or transcritical depending on the topography elevation. Using only four Wiener-Hermite basis functions, the model produces probability distributions comparable to those from a Monte Carlo reference simulation with 2,000 iterations while executing about 100 times faster. Accompanying model software and simulation data are openly available online.