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Positivity-preserving Well-balanced PAMPA Schemes with Global Flux quadrature for One-dimensional Shallow Water Models

Remi Abgrall, Yongle Liu, Mario Ricchiuto

TL;DR

We address the challenge of solving one-dimensional shallow-water balance laws with bottom topography, Manning friction, and Coriolis effects by developing a positivity-preserving, well-balanced PAMPA scheme that uses flux globalization. The core idea recasts the system as $\partial_t \mathbf{u} + \partial_x \mathbf{G}(\mathbf{u},x) = 0$, with a global flux $\mathbf{G}$ defined by the flux minus the source integral and evaluated via Gauss–Lobatto quadrature, enabling exact still-water preservation and super-convergence toward moving and geostrophic equilibria. A monolithic convex-limiting strategy blends high-order PAMPA fluxes with first-order Lax–Friedrichs-type fluxes to ensure positivity and WB while suppressing oscillations, and the scheme remains fully local and compact. Extensive numerical experiments demonstrate high-order accuracy in smooth regions, robust WB and PP properties for wet-dry fronts, and accurate capture of perturbations around equilibria, underscoring significant practical impact for geophysical and hydraulic applications.

Abstract

We present a novel hydrostatic and non-hydrostatic equilibria preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) method for solving the one-dimensional hyperbolic balance laws, with applications to the shallow water models including the Saint--Venant system with the Manning friction term and rotating shallow water equations. The idea is based on a global flux quadrature formulation, in which the discretization of the source terms is obtained from the derivative of and additional flux function computed via high order quadrature of the source term. The reformulated system is quasi-conservative with global integral terms computed using Gauss--Lobatto quadrature nodes. The resulting method is capable of preserving a large family of smooth moving equilibria: supercritical and subcritical flows, in a super-convergent manner. We also show that, by an appropriate quadrature strategy for the source, we can exactly preserve the still water states. Moreover, to guarantee the positivity of water depth and eliminate the spurious oscillations near shocks, we blend the high-order PAMPA schemes with the first order local Lax--Friedrichs schemes using the method developed in [R. Abgrall, M. Jiao, Y. Liu, and K. Wu, arXiv preprint arXiv:2410.14292, 2024]. The first-order schemes are designed to preserve the still water equilibria and positivity of water height, as well as to deal with wet-dry fronts. Extensive numerical experiments are tested to validate the advantages and robustness of the proposed scheme.

Positivity-preserving Well-balanced PAMPA Schemes with Global Flux quadrature for One-dimensional Shallow Water Models

TL;DR

We address the challenge of solving one-dimensional shallow-water balance laws with bottom topography, Manning friction, and Coriolis effects by developing a positivity-preserving, well-balanced PAMPA scheme that uses flux globalization. The core idea recasts the system as , with a global flux defined by the flux minus the source integral and evaluated via Gauss–Lobatto quadrature, enabling exact still-water preservation and super-convergence toward moving and geostrophic equilibria. A monolithic convex-limiting strategy blends high-order PAMPA fluxes with first-order Lax–Friedrichs-type fluxes to ensure positivity and WB while suppressing oscillations, and the scheme remains fully local and compact. Extensive numerical experiments demonstrate high-order accuracy in smooth regions, robust WB and PP properties for wet-dry fronts, and accurate capture of perturbations around equilibria, underscoring significant practical impact for geophysical and hydraulic applications.

Abstract

We present a novel hydrostatic and non-hydrostatic equilibria preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) method for solving the one-dimensional hyperbolic balance laws, with applications to the shallow water models including the Saint--Venant system with the Manning friction term and rotating shallow water equations. The idea is based on a global flux quadrature formulation, in which the discretization of the source terms is obtained from the derivative of and additional flux function computed via high order quadrature of the source term. The reformulated system is quasi-conservative with global integral terms computed using Gauss--Lobatto quadrature nodes. The resulting method is capable of preserving a large family of smooth moving equilibria: supercritical and subcritical flows, in a super-convergent manner. We also show that, by an appropriate quadrature strategy for the source, we can exactly preserve the still water states. Moreover, to guarantee the positivity of water depth and eliminate the spurious oscillations near shocks, we blend the high-order PAMPA schemes with the first order local Lax--Friedrichs schemes using the method developed in [R. Abgrall, M. Jiao, Y. Liu, and K. Wu, arXiv preprint arXiv:2410.14292, 2024]. The first-order schemes are designed to preserve the still water equilibria and positivity of water height, as well as to deal with wet-dry fronts. Extensive numerical experiments are tested to validate the advantages and robustness of the proposed scheme.

Paper Structure

This paper contains 14 sections, 6 theorems, 102 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Numerical scheme
  3. PP WB update for cell averages
  4. Low order fluxes
  5. High order fluxes
  6. PP WB update for point values
  7. Low order residuals
  8. High order residuals
  9. Convex combination coefficients
  10. Suppress spurious oscillations
  11. Positivity-preserving and well-balanced properties
  12. Numerical examples
  13. Conclusion
  14. Consistency of sub-cell LobattoIII collocation method

Key Result

Proposition 2.1

The implicit sub-cell LobattoIII RK me-thod defined by eq:scLIII1-eq:scLIII2 is fourth order accurate for smooth enough solutions.

Figures (15)

  • Figure 1: Sketch of the cut-off function $\mathcal{H}(\phi)$. \newlabelH_cutoff0
  • Figure 1: Example 2: Time snapshots of the small perturbation in $h$. \newlabelEx2_dh0
  • Figure 2: Example 3: Convergent solutions for supercritical flow. The deviation parameter $\epsilon =10^{-12}$. \newlabelEx3a0
  • Figure 3: Example 3: Convergent solutions for subcritical flow. The deviation parameter $\epsilon =10^{-12}$. \newlabelEx3b0
  • Figure 4: Example 3: logarithmic plots for errors in water depth versus number of cells. \newlabelEx30
  • ...and 10 more figures

Theorems & Definitions (17)

  • Proposition 2.1: sc-LIII consistency \newlabelpp:sc-lobattoIII0
  • Proof 1
  • Proposition 2.2: Compactness \newlabelcompactness0
  • Proof 2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1: Positivity-preserving of the first-order schemes
  • Proof 3
  • Proposition 3.2: Preserving still water of the first-order schemes
  • Proof 4
  • ...and 7 more