A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere

Abstract

In a previous article [J. Comp. Phys. $357$ (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.

References in corpus (11)

• Finite element exterior calculus: from Hodge theory to numerical stability
• Efficient Assembly of H(div) and H(curl) Conforming Finite Elements
• Mixed finite elements for numerical weather prediction
• A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations
• Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms
• Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements
• Mixed Mimetic Spectral Element Method for Stokes Flow: A Pointwise Divergence-Free Solution
• Mimetic framework on curvilinear quadrilaterals of arbitrary order
• Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
• Finite element differential forms on curvilinear cubic meshes and their approximation properties
• Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

Cited by in corpus (6)

• Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations
• Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere
• A Mixed Mimetic Spectral Element Model of the 3D Compressible Euler Equations on the Cubed Sphere
• Petrov-Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems
• An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics
• A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains