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A Well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) Method for Shallow Water Equations with Horizontal Temperature Gradients

Yongle Liu

TL;DR

The paper tackles solving the 2-D shallow water equations with horizontal temperature gradients on unstructured triangular meshes by introducing a well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) method. It combines a conservative update for cell averages with a non-conservative evolution of edge-based point values expressed in pressure-momentum-temperature variables, enabling exact preservation of both lake-at-rest and isobaric equilibria. The authors develop a third-order WB PAMPA scheme with an adaptive quadrature strategy and a MOOD-based nonlinear limiter to ensure positivity and suppress oscillations, while offering a compact stencil and reduced DoFs compared to traditional high-order methods. Numerical tests demonstrate high-order accuracy, robust handling of discontinuities, and faithful equilibrium preservation, confirming the method's potential for geophysical flows with temperature variations and paving the way for higher-order and more complex thermo-geophysical models.

Abstract

In this paper, we develop a novel well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) numerical method for solving the two-dimensional shallow water equations with temperature gradients on unstructured triangular meshes. The proposed PAMPA method use a globally continuous representation of the variables, with degree of freedoms (DoFs) consisting of point values on the edges and average values within each triangular element. The update of cell averages is carried out using a conservative form of the partial differential equations (PDEs), while the update of point values -- unconstrained by local conservation -- follows a non-conservative formulation. The powerful PAMPA framework offers great flexibility in the choice of variables for the non-conservative form, including conservative variables, primitive variables, and other possible sets of variables. In order to preserve a wider class of steady-state solutions, we introduce pressure-momentum-temperature variables instead of using the standard conservative or primitive ones. By utilizing these new variables and the associated non-conservative form, along with adopting suitable Gaussian quadrature rules in the discretization of conservative form, we prove that this new class of schemes is well-balanced for both ``lake at rest'' and isobaric steady states. We validate the performance of the proposed well-balanced PAMPA method through a series of numerical experiments, demonstrating their high-order accuracy, well-balancedness, and robustness.

A Well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) Method for Shallow Water Equations with Horizontal Temperature Gradients

TL;DR

The paper tackles solving the 2-D shallow water equations with horizontal temperature gradients on unstructured triangular meshes by introducing a well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) method. It combines a conservative update for cell averages with a non-conservative evolution of edge-based point values expressed in pressure-momentum-temperature variables, enabling exact preservation of both lake-at-rest and isobaric equilibria. The authors develop a third-order WB PAMPA scheme with an adaptive quadrature strategy and a MOOD-based nonlinear limiter to ensure positivity and suppress oscillations, while offering a compact stencil and reduced DoFs compared to traditional high-order methods. Numerical tests demonstrate high-order accuracy, robust handling of discontinuities, and faithful equilibrium preservation, confirming the method's potential for geophysical flows with temperature variations and paving the way for higher-order and more complex thermo-geophysical models.

Abstract

In this paper, we develop a novel well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) numerical method for solving the two-dimensional shallow water equations with temperature gradients on unstructured triangular meshes. The proposed PAMPA method use a globally continuous representation of the variables, with degree of freedoms (DoFs) consisting of point values on the edges and average values within each triangular element. The update of cell averages is carried out using a conservative form of the partial differential equations (PDEs), while the update of point values -- unconstrained by local conservation -- follows a non-conservative formulation. The powerful PAMPA framework offers great flexibility in the choice of variables for the non-conservative form, including conservative variables, primitive variables, and other possible sets of variables. In order to preserve a wider class of steady-state solutions, we introduce pressure-momentum-temperature variables instead of using the standard conservative or primitive ones. By utilizing these new variables and the associated non-conservative form, along with adopting suitable Gaussian quadrature rules in the discretization of conservative form, we prove that this new class of schemes is well-balanced for both ``lake at rest'' and isobaric steady states. We validate the performance of the proposed well-balanced PAMPA method through a series of numerical experiments, demonstrating their high-order accuracy, well-balancedness, and robustness.
Paper Structure (11 sections, 1 theorem, 64 equations, 16 figures, 6 tables, 1 algorithm)

This paper contains 11 sections, 1 theorem, 64 equations, 16 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Well-balanced third-order PAMPA schemes
  3. Approximation space
  4. Semi-discrete scheme for the update of cell average
  5. Semi-discrete schemes for the update of point value
  6. Summary of the third-order well-balanced schemes
  7. Non-linear stabilization and low-order schemes
  8. First-order scheme for cell average
  9. First-order scheme for point value
  10. Numerical examples
  11. Conclusion

Key Result

Proposition 2.6

\newlabelprop:WB0 The third-order WB PAMPA methods for the shallow water equations with temperature gradients, as described above, maintain the WB property for the steady states "lake at rest" 1.2a and isobaric 1.2b. That is, if the discrete data satisfy "lake at rest": or isobaric: then 2.6 and 2.12 reduce to

Figures (16)

  • Figure 1: Quadratic triangular element. \newlabelp2_ele0
  • Figure 1: Geometry for the first-order scheme. \newlabelp2_ele_subcell0
  • Figure 1: Example 2: Cell averages and point values of water depth at $t=5$. \newlabelEx2_h0
  • Figure 2: Mesh notation and the (extended) neighbourhood areas of an element $E$. \newlabelneighbour_cell0
  • Figure 2: Example 2: Cell averages and point values of water depth at $t=0.39$ (top row) and $t=1$ (bottom row) computed by the WB PAMPA scheme over non-flat bottom topography. \newlabelEx2_h10
  • ...and 11 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proof 1
  • Remark 2.7