Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Two-Dimensional Shallow Water Equations

TL;DR

This paper introduces a novel stochastic Galerkin method for two-dimensional shallow water equations that preserves hyperbolicity and well-balanced properties, ensuring accurate and robust simulations under uncertainty.

Contribution

It develops a hyperbolicity-preserving stochastic Galerkin formulation with a new scheme that maintains the physical properties of the shallow water system under randomness.

Findings

The method preserves hyperbolicity under certain positivity conditions.

The scheme is well-balanced and robust in numerical tests.

Numerical experiments demonstrate high accuracy and stability.

Abstract

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.