Energy Stable and Structure-Preserving Algorithms for the Stochastic Galerkin System of 2D Shallow Water Equations

TL;DR

This paper develops energy stable, structure-preserving finite volume algorithms for the stochastic Galerkin system of 2D shallow water equations, effectively handling uncertainty while maintaining stability and accuracy.

Contribution

It introduces novel structure-preserving, energy stable, second-order finite volume schemes for the stochastic Galerkin formulation of 2D shallow water equations, incorporating entropy flux and hyperbolicity preservation.

Findings

Algorithms demonstrate robustness in numerical experiments.

Methods effectively handle uncertainty in shallow water models.

Schemes are energy conservative and well-balanced.

Abstract

Shallow water equations (SWE) are fundamental nonlinear hyperbolic PDE-based models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. Therefore, stable and accurate numerical methods for SWE are needed. Although some algorithms are well studied for deterministic SWE, more effort should be devoted to handling the SWE with uncertainty. In this paper, we incorporate uncertainty through a stochastic Galerkin (SG) framework, and building on an existing hyperbolicity-preserving SG formulation for 2D SWE, we construct the corresponding entropy flux pair, and develop structure-preserving, well-balanced, second-order energy conservative and energy stable finite volume schemes for the SG formulation of the two-dimensional shallow water system. We demonstrate the efficacy, applicability, and robustness of these structure-preserving algorithms through several challenging numerical experiments.