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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.01199 |
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| _version_ | 1866915906431483904 |
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| author | Öffner, Philipp Pettersson, Per Winters, Andrew R. |
| author_facet | Öffner, Philipp Pettersson, Per Winters, Andrew R. |
| contents | Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. %In particular, Bender and Öffner 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01199 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension Öffner, Philipp Pettersson, Per Winters, Andrew R. Numerical Analysis Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. %In particular, Bender and Öffner 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. |
| title | A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.01199 |