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An entropy stable discontinuous Galerkin method for the spherical thermal shallow water equations

Kieran Ricardo, Kenneth Duru, David Lee

TL;DR

This work tackles stable long-time simulation of the rotating thermal shallow water equations (TRSW) on curved, spherical geometries by developing an entropy-stable discontinuous Galerkin (DG) method. It introduces a split-form of TRSW that avoids discrete chain-rule demands, derives a convex buoyancy-variance entropy, and designs novel entropy-stable and energy-stable fluxes within a curvilinear SBP-DG framework. The authors prove discrete conservation of mass, buoyancy, energy, vorticity, and buoyancy-variance entropy, and demonstrate semi-discrete entropy stability under appropriate flux choices via SBP properties. Numerical experiments on a cubed-sphere mesh validate the theory, showing robustness in well-developed geostrophic turbulence without artificial dissipation, and highlighting the method’s potential for scalable, accurate spherical atmospheric simulations. The study lays the groundwork for extending entropy-stable DG methods to related geophysical systems, including the Euler equations, in curvilinear geometries.

Abstract

We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.

An entropy stable discontinuous Galerkin method for the spherical thermal shallow water equations

TL;DR

This work tackles stable long-time simulation of the rotating thermal shallow water equations (TRSW) on curved, spherical geometries by developing an entropy-stable discontinuous Galerkin (DG) method. It introduces a split-form of TRSW that avoids discrete chain-rule demands, derives a convex buoyancy-variance entropy, and designs novel entropy-stable and energy-stable fluxes within a curvilinear SBP-DG framework. The authors prove discrete conservation of mass, buoyancy, energy, vorticity, and buoyancy-variance entropy, and demonstrate semi-discrete entropy stability under appropriate flux choices via SBP properties. Numerical experiments on a cubed-sphere mesh validate the theory, showing robustness in well-developed geostrophic turbulence without artificial dissipation, and highlighting the method’s potential for scalable, accurate spherical atmospheric simulations. The study lays the groundwork for extending entropy-stable DG methods to related geophysical systems, including the Euler equations, in curvilinear geometries.

Abstract

We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.
Paper Structure (26 sections, 11 theorems, 98 equations, 8 figures)

This paper contains 26 sections, 11 theorems, 98 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Continuous thermal shallow water equations
  3. Vector invariant equations
  4. Conservation of mass and mass-weighted buoyancy
  5. Conservation of total vorticity
  6. Conservation of energy
  7. Entropy
  8. Split form
  9. The DG method
  10. Spaces
  11. Operators
  12. Integrals
  13. Discrete formulation
  14. Semi-discrete conservation and numerical stability
  15. Conservation of mass and mass-weighted buoyancy
  16. ...and 11 more sections

Key Result

Theorem 2.1

Consider the TRSW eq:momentum--eq:buoyancy on a 2D manifold $\Omega$ embedded in $R^3$ with periodic boundary conditions. At time $t \ge 0$, let the total mass be denoted by $M(t) = \int_\Omega{hd\Omega}$ and the total mass-weighted buoyancy denoted by $S(t) = \int_\Omega{hbd\Omega}$. We have

Figures (8)

  • Figure 1: Steady state test case $L^2$ errors for the conservative and dissaptive methods at various resolutions. The resolution is taken as the average nodal spacing and results are shown at 5 days.
  • Figure 2: Relative energy and entropy conservation error at day 10 of the modified Galewsky test case for the conservative scheme with varying timesteps and $6\times5\times 5$ third order elements.
  • Figure 3: Conservation errors for modified Galewsky test case with the dissipative and conservative methods. A resolution of $6\times64\times64$ third order elements and a timestep of $0.8$ CFL was used.
  • Figure 4: Relative vorticity for the modified Galewsky test case at day 7 for the conservative (left), and dissipative methods (right). A resolution of $6\times64\times64$ third order elements and a timestep of $0.8$ CFL was used.
  • Figure 5: Buoyancy for the modified Galewsky test case at day 7 for the conservative (left), and dissipative methods (right). A resolution of $6\times64\times64$ third order elements and a timestep of $0.8$ CFL was used.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 15 more