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

## TL;DR

This paper develops a stochastic Galerkin finite volume model for shallow water flows over uncertain topography, ensuring well-balanced solutions and significantly reducing computational cost compared to Monte Carlo methods.

## Contribution

It introduces a low-order Wiener-Hermite Polynomial Chaos expansion with a finite volume approach that guarantees well-balancing and handles non-Gaussian uncertainties.

## Key findings

- Model preserves lake-at-rest condition analytically and numerically.
- Achieves comparable probability distributions to Monte Carlo with 100 times less computation.
- Successfully models flows with discontinuous and non-Gaussian topography uncertainties.

## Abstract

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 modelling 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 2000 iterations, while executing about 100 times faster. Accompanying model software and simulation data is openly available online.