A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations

Abstract

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured grids. The essential ingredients to achieve this are: (i) a velocity-vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular grids.

Toward free-surface flow simulations with correct energy evolution: An isogeometric level-set approach with monolithic time-integration

“…This is in contrast to the single fluid case where it does vanish, see e.g. [25,30,31]. The difference lies in the presence of the density.…”

Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

“…The scheme has an equivalent formulation where one discards the potential vorticity -velocity relationship (which is then a consequence of the subsequent equations) in favour of a potential vorticity evolution equation everywhere in the domain. This becomes reminiscent of the energy-enstrophy mimetic spectral element conserving formulation of Palha and Gerritsma (2017), in which both vorticity and velocity are maintained as prognostic variables and a timestaggered discretisation leads to efficient implementation with energy and enstrophy conservation, although in that case the correspondence between velocity and vorticity is not guaranteed by the timestepping scheme. In our numerical experiments we used an implicit energy-conserving timestepping scheme, which preserves the correspondence between velocity and vorticity, but does not preserve enstrophy exactly.…”

An Accurate Finite Element Method for the Numerical Solution of Isothermal and Incompressible Flow of Viscous Fluid

“…In [2] balance equations of mass, momentum, angular momentum, and energy are used for performing 3D computations without numerical parameters; however, the method already uses the energy equation such that generalization for the non-isothermal case seems to be quite difficult. In [54] vorticity is introduced as an independent term ensuring that the balance of moment of momentum is satisfied, in 2D numerical solutions are performed without necessitating any (numerical) parameters. In [18] different strategies are performed for establishing 3D simulations.…”