A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere
This work provides a geometrically flexible numerical method for geophysical fluid dynamics that preserves mimetic properties on non-affine grids.
The authors extended the mixed mimetic spectral element method to solve rotating shallow water equations on a cubed sphere, achieving conservation of mass, vorticity, and energy with spectral convergence.
In a previous article [J. Comp. Phys. $\mathbf{357}$ (2018) 282-304], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly varying, non-affine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in $H(\mathrm{rot})$, $H(\mathrm{div})$ and $L_2$. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the generalized Piola transformation used to construct the metric terms for the physical field quantities.