Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for two-layer shallow water equations

Jiahui Zhang, Yinhua Xia, Yan Xu

Abstract

This paper introduces well-balanced path-conservative discontinuous Galerkin (DG) methods for two-layer shallow water equations, ensuring exactness for both still water and moving water equilibrium steady states. The approach involves approximating the equilibrium variables within the DG piecewise polynomial space, while expressing the DG scheme in the form of path-conservative schemes. To robustly handle the nonconservative products governing momentum exchange between the layers, we incorporate the theory of Dal Maso, LeFloch, and Murat (DLM) within the DG method. Additionally, linear segment paths connecting the equilibrium functions are chosen to guarantee the well-balanced property of the resulting scheme. The simple ``lake-at-rest" steady state is naturally satisfied without any modification, while a specialized treatment of the numerical flux is crucial for preserving the moving water steady state. Extensive numerical examples in one and two dimensions validate the exact equilibrium preservation of the steady state solutions and demonstrate its high-order accuracy. The performance of the method and high-resolution results further underscore its potential as a robust approach for nonconservative hyperbolic balance laws.

Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for two-layer shallow water equations

Abstract

This paper introduces well-balanced path-conservative discontinuous Galerkin (DG) methods for two-layer shallow water equations, ensuring exactness for both still water and moving water equilibrium steady states. The approach involves approximating the equilibrium variables within the DG piecewise polynomial space, while expressing the DG scheme in the form of path-conservative schemes. To robustly handle the nonconservative products governing momentum exchange between the layers, we incorporate the theory of Dal Maso, LeFloch, and Murat (DLM) within the DG method. Additionally, linear segment paths connecting the equilibrium functions are chosen to guarantee the well-balanced property of the resulting scheme. The simple ``lake-at-rest" steady state is naturally satisfied without any modification, while a specialized treatment of the numerical flux is crucial for preserving the moving water steady state. Extensive numerical examples in one and two dimensions validate the exact equilibrium preservation of the steady state solutions and demonstrate its high-order accuracy. The performance of the method and high-resolution results further underscore its potential as a robust approach for nonconservative hyperbolic balance laws.
Paper Structure (23 sections, 5 theorems, 116 equations, 15 figures, 7 tables)

This paper contains 23 sections, 5 theorems, 116 equations, 15 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Model description and DLM theory
  3. Two-layer system
  4. DLM theory
  5. Still water equilibria preserving PCDG scheme
  6. Notations
  7. One-dimensional PCDG scheme for still water equilibrium
  8. Two-dimensional PCDG scheme for still water equilibrium
  9. Well-balanced property
  10. Moving water equilibria preserving PCDG scheme
  11. Conservative and equilibrium variables
  12. The semi-discrete PCDG scheme
  13. PCDG scheme
  14. Well-balanced treatment
  15. The fully-discrete PCDG scheme
  16. ...and 8 more sections

Key Result

Theorem 2.1

(Dal Maso, LeFloch, and Murat (DLMdal1995definition)) Let $\boldsymbol {u}: [a,b]\rightarrow \mathbb{R}^l$ be a function of bounded variation and $\mathcal{A}: \mathbb{R}^l \rightarrow \mathbb{R}^{l\times l}$ be a smooth locally bounded matrix-valued function. Then, there exists a unique real-valued The Borel measure $\mu$ is called nonconservative product and can be written as $[\mathcal{A}(\bold

Figures (15)

  • Figure 5.1: Example \ref{['2lswe:perstill']}: Numerical solutions of the water surface $h_1+w$ with the continuous bottom topography $(\ref{['smo1D']})$ (left) and the discontinuous bottom topography $(\ref{['dis1D']})$ (right), using 200 and 3000 grid cells.
  • Figure 5.2: Example \ref{['pro1d']}: From left to right: Numerical solutions of upper layer $h_1$ zoomed at the interface area, water surface $h_1+w$ and velocity of the upper layer $u_1$ with initial conditions (\ref{['pro']}).
  • Figure 5.3: Example \ref{['pro1d']}: Numerical solutions of interface $w$ (left) and water surface $h_1+w$ (right) with initial conditions (\ref{['pro_large']}). Top: using 200 and 5000 grid cells; bottom: using CU, PCCU, PCDG-still, and PCDG-moving schemes for comparison.
  • Figure 5.4: Example \ref{['dam1d']}: Left: The discrete steady states for the water surface $h_1+w$ and interface $w$, with 200 cells; right: using CU, PCCU, PCDG-still schemes for comparison, with 2000 grid cells.
  • Figure 5.5: Example \ref{['iso1d']}: From top to bottom: Numerical solutions of the lower layer $h_2$ and water depth $h_1+h_2$, zoom-in of the lower layer $h_2$ and the water depth $h_1+h_2$. Left: using 200 and 2000 grid cells; right: using CU, PCCU, PCDG-still schemes for comparison.
  • ...and 10 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 4.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 14 more