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Energy Stable and Structure-Preserving Algorithms for the Stochastic Galerkin System of 2D Shallow Water Equations

Yekaterina Epshteyn, Akil Narayan, Yinqian Yu

TL;DR

This work addresses uncertainty in the two-dimensional shallow water equations by employing an intrusive stochastic Galerkin (SG) framework built on a hyperbolicity-preserving formulation. It derives an entropy flux pair for SG SWE and develops well-balanced, energy-conservative (EC) and energy-stable (ES) finite-volume schemes, including a second-order EC and first-/second-order ES methods, with Desingularization and hyperbolicity controls. The proposed schemes are validated through a battery of challenging tests, including stochastic bottom/topography and multi-dimensional random inputs, demonstrating energy dissipation behavior, robustness, and accurate capture of lake-at-rest states under uncertainty. The results have practical impact for reliable long-time simulations of geophysical flows with uncertain data, and the framework can be extended to higher-order methods and dry/wet interfaces. Key contributions include the SG entropy pair, EC/ES schemes with well-balanced properties, and a comprehensive numerical assessment of energy behavior under stochastic perturbations.

Abstract

Shallow water equations (SWE) are fundamental nonlinear hyperbolic PDE-based models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. Therefore, stable and accurate numerical methods for SWE are needed. Although some algorithms are well studied for deterministic SWE, more effort should be devoted to handling the SWE with uncertainty. In this paper, we incorporate uncertainty through a stochastic Galerkin (SG) framework, and building on an existing hyperbolicity-preserving SG formulation for 2D SWE, we construct the corresponding entropy flux pair, and develop structure-preserving, well-balanced, second-order energy conservative and energy stable finite volume schemes for the SG formulation of the two-dimensional shallow water system. We demonstrate the efficacy, applicability, and robustness of these structure-preserving algorithms through several challenging numerical experiments.

Energy Stable and Structure-Preserving Algorithms for the Stochastic Galerkin System of 2D Shallow Water Equations

TL;DR

This work addresses uncertainty in the two-dimensional shallow water equations by employing an intrusive stochastic Galerkin (SG) framework built on a hyperbolicity-preserving formulation. It derives an entropy flux pair for SG SWE and develops well-balanced, energy-conservative (EC) and energy-stable (ES) finite-volume schemes, including a second-order EC and first-/second-order ES methods, with Desingularization and hyperbolicity controls. The proposed schemes are validated through a battery of challenging tests, including stochastic bottom/topography and multi-dimensional random inputs, demonstrating energy dissipation behavior, robustness, and accurate capture of lake-at-rest states under uncertainty. The results have practical impact for reliable long-time simulations of geophysical flows with uncertain data, and the framework can be extended to higher-order methods and dry/wet interfaces. Key contributions include the SG entropy pair, EC/ES schemes with well-balanced properties, and a comprehensive numerical assessment of energy behavior under stochastic perturbations.

Abstract

Shallow water equations (SWE) are fundamental nonlinear hyperbolic PDE-based models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. Therefore, stable and accurate numerical methods for SWE are needed. Although some algorithms are well studied for deterministic SWE, more effort should be devoted to handling the SWE with uncertainty. In this paper, we incorporate uncertainty through a stochastic Galerkin (SG) framework, and building on an existing hyperbolicity-preserving SG formulation for 2D SWE, we construct the corresponding entropy flux pair, and develop structure-preserving, well-balanced, second-order energy conservative and energy stable finite volume schemes for the SG formulation of the two-dimensional shallow water system. We demonstrate the efficacy, applicability, and robustness of these structure-preserving algorithms through several challenging numerical experiments.

Paper Structure

This paper contains 35 sections, 13 theorems, 131 equations, 16 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions of this paper
  3. Stochastic Galerkin formulation of the two-dimensional shallow water equations
  4. Polynomial chaos expansion
  5. Hyperbolic-preserving stochastic Galerkin formulation for 2D shallow water equation
  6. An entropy flux pair for the 2D SG SWE
  7. Entropy flux pairs for the deterministic SWE
  8. An entropy flux pair for the 2D SG SWE
  9. Well-balanced energy conservative and energy stable schemes for the 2D SG SWE
  10. Energy conservative schemes
  11. An energy conservative scheme for the 2D SG SWE (EC)
  12. A first-order energy stable scheme (ES1)
  13. A second-order energy stable scheme (ES2)
  14. Algorithmic details and pseudocode
  15. Desingularization
  16. ...and 20 more sections

Key Result

Theorem 1

If the matrix $\mathcal{P}(\widehat{h})$ is strictly positive definite at every point $(x,y,t)$ in the computational spatial-temporal domain, then the SG formulation SGSWE is hyperbolic.

Figures (16)

  • Figure 1: Illustration of energy versus augmented energy that incorporates energy fluxes at the boundary. Top left: Bottom topography $B$ corresponding to \ref{['eq:augenergy-example']}. Top right: Mean of the water surface $w = h + B$ at terminal time $T = 0.07$. Bottom left: The standard relative energy $\frac{\bm{E}(t) - \bm{E}(0)}{\bm{E}(0)}$ increases in time for any accurate scheme due to influx of energy at the boundaries. Bottom right: Measuring the relative change in augmented energy $\frac{\widetilde{\bm{E}}(t) - \widetilde{\bm{E}}(0)}{\widetilde{\bm{E}}(0)}$ that offsets energy change due to boundary effects restores expected behavior for relative change of energy when using EC and ES schemes.
  • Figure 2: Results for \ref{['Sec5_1']}: Contours for the water surface of the reference solution of ES2. Left: mean. Right: standard deviation. $T=0.07$.
  • Figure 3: Results for \ref{['GShump_stochasticbottom']}. Mean of water surface. Disk-glyph over mean contours, where the radii of the disks indicate the magnitude of the standard deviation. Left: ES1, the maximum standard deviation is 4.6524e-04, 7.3933e-04, 3.5168e-04, 1.3633e-04, 1.0856e-04, respectively. Right: ES2, the maximum standard deviation is 1.0398e-03, 2.1739e-03, 8.9851e-04, 3.4970e-04, 2.9730e-04, respectively. $200\times 200$, $t = 0.6, 0.9, 1.2, 1.5, 1.8$.
  • Figure 4: Results for \ref{['GShump_stochasticbottom']}. Left: relative change in energy. Right: relative change in augmented energy.
  • Figure 5: Results for \ref{['Sec5_3']}. Mean of water surface. Disk-glyph over mean contours, where the radii of the disks indicate the magnitude of the standard deviation. Left: ES1, the maximum standard deviation is 1.8216e-03, 1.9375e-03, 1.5528e-03, 1.0162e-03, 8.2437e-04, respectively. Right: ES2, the maximum standard deviation is 2.6787e-03, 3.2736e-03, 2.5036e-03, 1.5027e-03, 1.3321e-03, respectively. $200\times 200$, $t = 0.6, 0.9, 1.2, 1.5, 1.8$.
  • ...and 11 more figures

Theorems & Definitions (18)

  • Theorem 1: Theorem 3.1 in dai2022hyperbolicity
  • Theorem 2
  • Lemma 1: Gradients of $\widehat{u}, \widehat{v}$
  • Lemma 2: Convexity of $E(\widehat{U})$
  • Lemma 3: Companion balance law
  • Definition 1: Well-Balanced SG SWE Property dai2022hyperbolicity
  • Lemma 4
  • Definition 2: Energy conservative and energy stable schemes
  • Theorem 3: Second-order EC well-balanced scheme
  • Lemma 5: Second-order truncation error
  • ...and 8 more