Conservative finite volume schemes for the quasi-geostrophic equation on coastal-conforming unstructured primal-dual meshes

TL;DR

This paper introduces finite volume schemes for the quasi-geostrophic equations on unstructured meshes, ensuring conservation properties and exploring boundary condition enforcement, with numerical validation on Voronoi-Delaunay meshes.

Contribution

It presents novel finite volume schemes for quasi-geostrophic equations on coastal-conforming unstructured meshes, including boundary condition strategies and conservation analysis.

Findings

Potential vorticity is conserved along fluid paths.

Potential enstrophy is conserved up to time truncation errors.

Numerical tests confirm the schemes' conservation properties.

Abstract

In this paper we propose finite volume schemes for solving the inviscid and viscous quasi-geostrophic equations on coastal-conforming unstructured primal-dual meshes. Several approaches for enforcing the boundary conditions are also explored along with these schemes. The pure transport part in these schemes are shown to conserve the potential vorticity along fluid paths in an averaged sense, and conserve the potential enstrophy up to the time truncation errors. Numerical tests based on the centroidal Voronoi-Delaunay meshes are performed to confirm these properties, and to distinguish the dynamical behaviors of these schemes. Finally some potential applications of these schemes in different situations are discussed.