Higher-order finite volume differential operators with selective upwinding on the icosahedral spherical grid

Abstract

The icosahedral hexagonal grid is a quasi-uniform discretization of the sphere, generated as the dual of a triangular grid derived by successively refining the faces of an icosahedron embedded in the sphere. This grid contains twelve pentagonal cells and a large number of hexagonal cells, and the bounded area ratio between the smallest and largest cells eliminates polar singularities.This structure, however, makes the derivation of high-order numerical methods more difficult than for grids with a more regular structure. This work progresses towards the goal of high-order operators for the dynamical core of a global atmospheric model by developing a nonstaggered finite volume method on this grid, featuring upwinding applied only when necessary for stability, using an edge-centered reconstruction that directly accommodates the grid's intrinsic curvature.These operators compute the cell-average divergence, curl, and gradient with accuracy between second and fourth order for test functions. When applied to shallow water test cases, the resulting method is between third and fourth order accurate for the transport equation and between second and third order accurate for the full shallow water system.