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Entropy-Stable Discontinuous Spectral-Element Methods for the Spherical Shallow Water Equations in Covariant Form

Tristan Montoya, Andrés M. Rueda-Ramírez, Gregor J. Gassner

TL;DR

The paper addresses constructing high-order, entropy-stable discretizations for the covariant rotating shallow water equations on curved manifolds, enabling robust dynamical-core-like simulations. It introduces a skew-symmetric covariant flux formulation discretized with SBP tensor-product spectral elements and flux differencing, yielding entropy-conservative and entropy-stable variants that either conserve or dissipate total energy while preserving the lake-at-rest balance for continuous bottoms. Theoretical results prove semi-discrete entropy balance without requiring discrete metric identities, and mass conservation holds for symmetric fluxes; numerical experiments on cubed-sphere grids demonstrate optimal convergence, long-time stability, and robustness against under-resolved vortical dynamics. This framework lays groundwork for next-generation atmospheric cores and suggests extensions to 3D nonhydrostatic models and other curved geometries in geophysical fluid dynamics and beyond.

Abstract

We introduce discontinuous spectral-element methods of arbitrary order that are well balanced, conservative of mass, and conservative or dissipative of total energy (i.e., a mathematical entropy function) for a covariant flux formulation of the rotating shallow water equations with variable bottom topography on curved manifolds such as the sphere. The proposed methods are based on a skew-symmetric splitting of the tensor divergence in covariant form, which we implement and analyze within a general flux-differencing framework using tensor-product summation-by-parts operators. Such schemes are proven to satisfy semi-discrete mass and energy conservation on general unstructured quadrilateral grids in addition to well balancing for arbitrary continuous bottom topographies, with energy dissipation resulting from a suitable choice of numerical interface flux. Furthermore, the proposed covariant formulation permits an analytical representation of the geometry and associated metric terms while satisfying the aforementioned entropy stability, conservation, and well-balancing properties without the need to approximate the metric terms so as to enforce discrete metric identities. Numerical experiments on cubed-sphere grids are presented in order to verify the schemes' structure-preservation properties as well as to assess their accuracy and robustness within the context of several standard test cases characteristic of idealized atmospheric flows. Our theoretical and numerical results support the further development of the proposed methodology towards a full dynamical core for numerical weather prediction and climate modelling, as well as broader applications to other hyperbolic and advection-dominated systems of partial differential equations on curved manifolds.

Entropy-Stable Discontinuous Spectral-Element Methods for the Spherical Shallow Water Equations in Covariant Form

TL;DR

The paper addresses constructing high-order, entropy-stable discretizations for the covariant rotating shallow water equations on curved manifolds, enabling robust dynamical-core-like simulations. It introduces a skew-symmetric covariant flux formulation discretized with SBP tensor-product spectral elements and flux differencing, yielding entropy-conservative and entropy-stable variants that either conserve or dissipate total energy while preserving the lake-at-rest balance for continuous bottoms. Theoretical results prove semi-discrete entropy balance without requiring discrete metric identities, and mass conservation holds for symmetric fluxes; numerical experiments on cubed-sphere grids demonstrate optimal convergence, long-time stability, and robustness against under-resolved vortical dynamics. This framework lays groundwork for next-generation atmospheric cores and suggests extensions to 3D nonhydrostatic models and other curved geometries in geophysical fluid dynamics and beyond.

Abstract

We introduce discontinuous spectral-element methods of arbitrary order that are well balanced, conservative of mass, and conservative or dissipative of total energy (i.e., a mathematical entropy function) for a covariant flux formulation of the rotating shallow water equations with variable bottom topography on curved manifolds such as the sphere. The proposed methods are based on a skew-symmetric splitting of the tensor divergence in covariant form, which we implement and analyze within a general flux-differencing framework using tensor-product summation-by-parts operators. Such schemes are proven to satisfy semi-discrete mass and energy conservation on general unstructured quadrilateral grids in addition to well balancing for arbitrary continuous bottom topographies, with energy dissipation resulting from a suitable choice of numerical interface flux. Furthermore, the proposed covariant formulation permits an analytical representation of the geometry and associated metric terms while satisfying the aforementioned entropy stability, conservation, and well-balancing properties without the need to approximate the metric terms so as to enforce discrete metric identities. Numerical experiments on cubed-sphere grids are presented in order to verify the schemes' structure-preservation properties as well as to assess their accuracy and robustness within the context of several standard test cases characteristic of idealized atmospheric flows. Our theoretical and numerical results support the further development of the proposed methodology towards a full dynamical core for numerical weather prediction and climate modelling, as well as broader applications to other hyperbolic and advection-dominated systems of partial differential equations on curved manifolds.

Paper Structure

This paper contains 22 sections, 8 theorems, 92 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Shallow water equations on manifolds
  3. Vector and tensor notation
  4. Shallow water equations in covariant form
  5. Entropy analysis
  6. Skew-symmetric formulation
  7. Discontinuous spectral-element methods
  8. Spherical quadrilateral mapping
  9. Spectral-element approximation
  10. Summation-by-parts operators
  11. Flux-differencing discretization
  12. Entropy-stable covariant formulation
  13. Entropy-conservative and entropy-stable two-point flux functions
  14. Mass conservation
  15. Entropy conservation and entropy stability
  16. ...and 7 more sections

Key Result

Lemma 1

The mapping from $\boldsymbol{u}$ to $\eta$ is strictly convex when $h > 0$ and $\boldsymbol{G}$ is SPD.

Figures (10)

  • Figure 1: Illustration of the surface $S$, reference domain $D$, and physical element $E$
  • Figure 2: Convergence results for the unsteady solid-body rotation problem at $t = 5\text{ days}$
  • Figure 3: Error growth, mass conservation, and total energy (i.e., mathematical entropy) conservation or dissipation for the isolated mountain problem with $(N,M) = (3,20)$
  • Figure 4: Relative vorticity field ($\mathrm{s}^{-1}$) at $t = 7 \text{ days}$ for the isolated mountain problem with $V = 20 \text{ m/s}$, where topography contours at $500 \text{ m}$, $1000 \text{ m}$, and $1500 \text{ m}$ are shown in black
  • Figure 5: Error growth and total energy (i.e., mathematical entropy) conservation or dissipation for the unperturbed barotropic instability problem
  • ...and 5 more figures

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Lemma 3
  • proof
  • Remark 4
  • ...and 16 more