Conservative transport schemes for spherical geodesic grids: High-order flux operators for ODE-based time integration

These operators are generalizations of 3rd- and 4th-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. 2D least-squares-fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading order truncation error terms of the standard 2nd-order centered formulation.

Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge-Kutta stage within a given time step. The 3rd- and 4th-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature we have examined.