A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension

TL;DR

This paper unifies two approaches to stochastic Galerkin shallow water equations, extends high-order entropy conservative DG methods to 2D, and verifies their stability and well-balanced properties through simulations.

Contribution

It provides a unified framework connecting two existing methods and develops 2D high-order entropy conservative DG schemes for stochastic shallow water equations.

Findings

The two formulations align under specific conditions.

The 2D high-order DG scheme is entropy stable and well-balanced.

Numerical simulations confirm the theoretical properties.

Abstract

Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. %In particular, Bender and \"Offner developed entropy-conservative discontinuous Galerkin (DG) methods to solve the stochastic shallow water equations in an stochastic Galerkin framework using Roe variable transformation, while Dai, Epshteyn and collaborators proposed second-order, energy-stable and well-balanced schemes for the same class of problems with a specific projection step used inside the Galerkin projection together with high-order quadrature rules and a time-step restriction. In this paper, we provide a comprehensive comparison of the two methodologies mentioned, focusing on their theoretical properties and practical implementation aspects. We highlight shared foundational concepts and key differences of both approaches, with a particular focus on the selection of basis functions in the stochastic domain. As a highlight, we show that under specific conditions, the two formulations align, offering a unified framework that connects these distinct approaches. From our theoretical findings, we extend the development of high-order entropy conservative DG methods for the one-dimensional stochastic Galerkin shallow equations to two space dimensions; constructing entropy conservative two-point fluxes via primitive variables instead of entropy variables and applying it in our high-order DG setting. In numerical simulations, we verify and support our theoretical findings of a well-balanced and entropy-stable DG scheme which can be used to solve geophyiscal fluid flows with uncertainty.