Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations

TL;DR

This paper develops energy stable and structure-preserving numerical schemes for the stochastic Galerkin shallow water equations, enabling accurate and stable simulations of water flows with uncertainty.

Contribution

It introduces entropy-based schemes that preserve physical properties and stability for stochastic shallow water models, advancing numerical methods for uncertain hyperbolic PDEs.

Findings

Schemes are entropy-entropy flux compatible.

Methods demonstrate stability and accuracy in numerical tests.

Preserve positivity and energy conservation in simulations.

Abstract

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.