Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as quadratic moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence

A FIC-based stabilized finite element formulation for turbulent flows

We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework. In comparison to existing FIC approaches for fluids, this formulation involves a new term in the momentum equation, which introduces non-isotropic dissipation in the direction of velocity gradients. We also follow a new approach to the derivation of the stabilized mass equation, inspired by recent developments for quasi-incompressible flows. The presented FIC-FEM formulation is used to simulate turbulent flows, using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation.Peer ReviewedPostprint (author's final draft