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Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations

Kieran Ricardo, David Lee, Kenneth Duru

TL;DR

This work develops an entropy-stable discontinuous Galerkin spectral element method for the vector-invariant rotating shallow water equations on the sphere. By leveraging entropy-stable fluxes and a vector-invariant formulation, the method locally conserves mass and absolute vorticity, semi-discretely conserves energy, and preserves exact discrete linear geostrophic balance on curvilinear meshes. The approach is validated on a cubed-sphere grid, showing stability without artificial dissipation and accurate representation of geostrophic turbulence, with adjustable dissipation via flux parameters to control high-frequency noise. The results provide a scalable, physically faithful discretisation suitable for long-range atmospheric simulations and offer a foundation for extending to more complex shallow water and thermodynamic models.

Abstract

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.

Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations

TL;DR

This work develops an entropy-stable discontinuous Galerkin spectral element method for the vector-invariant rotating shallow water equations on the sphere. By leveraging entropy-stable fluxes and a vector-invariant formulation, the method locally conserves mass and absolute vorticity, semi-discretely conserves energy, and preserves exact discrete linear geostrophic balance on curvilinear meshes. The approach is validated on a cubed-sphere grid, showing stability without artificial dissipation and accurate representation of geostrophic turbulence, with adjustable dissipation via flux parameters to control high-frequency noise. The results provide a scalable, physically faithful discretisation suitable for long-range atmospheric simulations and offer a foundation for extending to more complex shallow water and thermodynamic models.

Abstract

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.
Paper Structure (32 sections, 90 equations, 7 figures)

This paper contains 32 sections, 90 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Continuous shallow water equations
  3. Vector invariant equations
  4. Conservation of mass, absolute vorticity, and energy
  5. Linear geostrophic balance
  6. Entropy
  7. Spatial discretisation
  8. Function spaces
  9. Covariant and contravariant base vectors
  10. Discrete inner product and boundary integrals
  11. Discrete operators
  12. Properties of function spaces
  13. Element local summation by parts
  14. Divergence curl cancellation
  15. Continuity of interpolants
  16. ...and 17 more sections

Figures (7)

  • Figure 1: L2 difference from discrete initial conditions for the linear geostrophic balance test case.
  • Figure 2: $L^2$ error for the energy dissipating and energy conserving methods applied to Williamson test case 2 at day 5.
  • Figure 3: Test case 5 depth field at day 15 for the dissipative method at a resolution of $6\times32\times32$ elements. The contour interval is 50m.
  • Figure 4: Test case 6 depth field at day 14 for the dissipative method at a resolution of $6\times16\times16$ elements. The contour interval is 200m.
  • Figure 5: Energy conservation error for the centred flux scheme at low resolution for the Galewsky test case
  • ...and 2 more figures