Conservative numerical schemes with optimal dispersive wave relations -- Part I. Derivations and analyses

TL;DR

This paper develops energy- and enstrophy-conserving numerical schemes for inviscid shallow water flows, maintaining optimal dispersive wave relations on unstructured meshes by approximating Hamiltonian formulations.

Contribution

It introduces novel discretizations that preserve key physical invariants and dispersive properties, extending the Z-grid scheme to unstructured meshes.

Findings

Schemes conserve energy and enstrophy due to skew-symmetry.

They operate effectively on unstructured orthogonal dual meshes.

Dispersive wave relations are optimal, matching Z-grid scheme.

Abstract

An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived, by approximating the Hamiltonian formulation, based on the Poisson brackets and the vorticity-divergence variables, of the inviscid shallow water flows. The conservation of the energy and/or enstrophy stems from skew-symmetry of the Poisson brackets, which is retained in the discrete approximations. These schemes operate on unstructured orthogonal dual meshes, over bounded or unbounded domains, and they are also shown to possess the same optimal dispersive wave relations as those of the Z-grid scheme.