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

TL;DR

This paper introduces a mimetic spectral element discretization for 2D incompressible Navier-Stokes equations that exactly conserves key physical quantities like mass, energy, enstrophy, and vorticity on unstructured grids, ensuring high fidelity in simulations.

Contribution

It develops a novel velocity-vorticity formulation with specialized function spaces and a conserving time integrator to achieve exact discrete conservation laws.

Findings

Exact conservation of mass, energy, enstrophy, and vorticity demonstrated

Numerical tests on irregular grids confirm theoretical properties

Method maintains physical invariants in the limit of vanishing dissipation

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.