A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension
This work provides a unified framework for solving geophysical fluid flows with uncertainty, but it is incremental as it builds on and extends prior methods.
The paper compares two existing stochastic Galerkin methods for shallow water equations, showing they unify under specific conditions, and extends a high-order entropy-conservative discontinuous Galerkin scheme from one to two dimensions, verifying its well-balanced and entropy-stable properties in numerical simulations.
Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. %In particular, Bender and Ãffner 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.