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

Philipp Öffner, Per Pettersson, Andrew R. Winters

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 27 sections, 6 theorems, 81 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Spectral expansions in random variables
  3. Stochastic Galerkin projection
  4. Properties of different bases
  5. Stochastic Galerkin shallow water equations
  6. Hyperbolic preserving SG-SW formulation
  7. SG-SW formulation via Roe variable transformation
  8. Relation between the different SG-SW formulations
  9. Structure preserving numerical methods
  10. Entropy stable and well-balanced discontinuous Galerkin for the SG-SW formulation
  11. Entropy conservative numerical two-point fluxes
  12. The SG-SW formulation in two space dimensions
  13. Numerical results
  14. One dimensional problems
  15. Well-balancing
  16. ...and 12 more sections

Key Result

Lemma 2.1

For any basis satisfying eq:basis_P1, the following is true for all $\hat{v},\hat{w} \in \mathbb{R}^{K}$ unless otherwise stated: $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: The first few Legendre polynomials (a), Haar wavelets (b), and piecewise quadratic multi-wavelets (c).
  • Figure 2: Legendre polynomial and Haar wavelet reconstruction of the nonsmooth functions $f_1$ (a), $f_2$ (b). The factors $(1+\xi)$ and $(1-\xi^2)$ can be exactly represented with $K=3$ Legendre polynomials, but the product $(1+\xi)(1-\xi^2)$ projected onto the same basis assumes negative values with non-zero probability (c).
  • Figure 3: Polynomials of degree $N=4$, $64$ elements. Evolution of the entropy for different number of wavelets using LLF type dissipation.
  • Figure 4: Polynomials of degree $N=4$, $64$ elements, and $8$ wavelets. (left) Initial condition. (right) Mean solution / bottom topography and quantiles at the final time $t_{\text{final}} = 0.65$
  • Figure 5: Polynomials of degree $N=4$, $64$ elements. Evolution of the entropy for different number of wavelets using LLF type dissipation.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • ...and 2 more