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arXiv · 2310.06229

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

Dihan Dai·Yekaterina Epshteyn·Akil Narayan
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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.

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Dihan Dai, Yekaterina Epshteyn, Akil Narayan. 2023-10-10. Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations. https://arxiv.org/abs/2310.06229

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