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A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations

Remi Abgrall, Yongle Liu

TL;DR

This work develops a novel class of positivity-preserving, well-balanced schemes for the 1-D Saint-Venant equations by simultaneously leveraging conservative and primitive formulations. By evolving cell averages with the conservative form and point values with the primitive form, and linking them through a third-order parabolic interpolant and equilibrium variables, the method exactly preserves both still-water and moving-water equilibria without nonlinear root-finding. A MOOD-based positivity mechanism ensures nonnegative water depth, while flux splitting and a WB source-term treatment maintain balance. Numerical experiments demonstrate high-order accuracy, robustness to discontinuous bathymetry, and competitive performance against existing WB schemes, with a framework extensible to other hyperbolic models.

Abstract

In this paper, we introduce a new approach for constructing robust well-balanced numerical methods for the one-dimensional Saint-Venant system with and without the Manning friction term. Following the idea presented in [R. Abgrall, Commun. Appl. Math. Comput. 5(2023), pp. 370-402], we first combine the conservative and non-conservative (primitive) formulations of the studied conservative hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point values and average values. The point values and average values will then be evolved by two different forms of PDEs: a conservative version of the cell averages and a possibly non-conservative one for the points. We show how to deal with both the conservative and non-conservative forms of PDEs in a well-balanced manner. The developed schemes are capable of exactly preserving both the still-water and moving-water equilibria. Compared with existing well-balanced methods, this new class of scheme is nonlinear-equations-solver-free. This makes the developed schemes less computationally costly and easier to extend to other models. We demonstrate the behavior of the proposed new scheme on several challenging examples.

A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations

TL;DR

This work develops a novel class of positivity-preserving, well-balanced schemes for the 1-D Saint-Venant equations by simultaneously leveraging conservative and primitive formulations. By evolving cell averages with the conservative form and point values with the primitive form, and linking them through a third-order parabolic interpolant and equilibrium variables, the method exactly preserves both still-water and moving-water equilibria without nonlinear root-finding. A MOOD-based positivity mechanism ensures nonnegative water depth, while flux splitting and a WB source-term treatment maintain balance. Numerical experiments demonstrate high-order accuracy, robustness to discontinuous bathymetry, and competitive performance against existing WB schemes, with a framework extensible to other hyperbolic models.

Abstract

In this paper, we introduce a new approach for constructing robust well-balanced numerical methods for the one-dimensional Saint-Venant system with and without the Manning friction term. Following the idea presented in [R. Abgrall, Commun. Appl. Math. Comput. 5(2023), pp. 370-402], we first combine the conservative and non-conservative (primitive) formulations of the studied conservative hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point values and average values. The point values and average values will then be evolved by two different forms of PDEs: a conservative version of the cell averages and a possibly non-conservative one for the points. We show how to deal with both the conservative and non-conservative forms of PDEs in a well-balanced manner. The developed schemes are capable of exactly preserving both the still-water and moving-water equilibria. Compared with existing well-balanced methods, this new class of scheme is nonlinear-equations-solver-free. This makes the developed schemes less computationally costly and easier to extend to other models. We demonstrate the behavior of the proposed new scheme on several challenging examples.
Paper Structure (11 sections, 2 theorems, 67 equations, 12 figures, 5 tables)

This paper contains 11 sections, 2 theorems, 67 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Schemes
  3. Positivity-Preserving
  4. Numerical Examples
  5. Conclusion
  6. A third-order parabolic interpolant
  7. Desingularization
  8. Numerical tests over discontinuous bathymetries
  9. Test 1
  10. Test 2
  11. Test 3

Key Result

Theorem 2.1

\newlabelthm20 If the solution $\bm U$ is in equilibrium (1.3) at all Gauss-Lobatto points $\{x_j,x_{j+\frac{1}{2}},x_{j+1}\}$, $\forall K_{j+\frac{1}{2}}$, namely, the discrete data $\{\bm U_j, \bm U_{j+\frac{1}{2}}, \bm U_{j+1}\}$ satisfy the following relations: then, the proposed schemes given by (2.5), (2.6), (2.23)--(sint) and (2.8), (rnew) are WB, provided that the right-hand sides of (2.

Figures (12)

  • Figure 1: Sketch of the proposed WB active flux like scheme stabilized by a MOOD loop. \newlabelmood0
  • Figure 1: Example 2 (small perturbation): Time snapshots of the difference $h-h_{\rm eq}$ computed by the WB AF and WB PCCU schemes using a coarse mesh with $100$ cells (top row) and a finer mesh with $300$ cells (bottom row). \newlabelEx2_fig10
  • Figure 1: Small perturbation: same as in Figure \ref{['fig2']} but over discontinuous bathymetry (\ref{['zz2']}). \newlabelfig2a0
  • Figure 2: Example 3: The difference between the obtained and the steady-state cell averages of water depth computed by both the WB AF and WB PPCU schemes using uniform cells with mesh size $\Delta x=0.25$ (top row) and $\Delta x=0.025$ (bottom row) for Cases (a, left column), (b, middle column) and (c, right column). \newlabelfig20
  • Figure 2: Convergent solutions: Same as in Figure \ref{['fig4']} but over discontinuous bathemetry \ref{['zz2']}. \newlabelfig4db0
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 2.1: Well-balanced for general steady state
  • Theorem 2.2
  • Remark 3.1
  • Remark 3.2