On mass conservation in high-order high-resolution rigorous remapping schemes on the sphere

It is the purpose of this short article to analyze mass conservation in high-order rigorous remapping schemes, which contrary to flux-based methods, relies on elaborate integral constraints over overlap areas and reconstruction functions. For applications on thesphere these integral constraints may be violated primarily as a result of inexact or ill-conditioned integration and the authors propose a generic, local, and multitracer efficient method that guarantees that the integral constraints are satisfied in discrete space irrespective of the accuracy of the numerical integration method and slight inaccuracies in the computation of overlap areas. The authors refer to this method as enforcement of consistency as it is based on integral constraints valid in continuous space. The consistency enforcement method is illustrated in idealized transport tests with the Conservative Semi-Lagrangian Multitracer scheme (CSLAM) in the High Order Method Modeling Environment(HOMME) where the analytic integrals, which were found to be ill conditioned at certain resolutions and flow conditions, have been replaced with robust quadrature. This violates mass conservation; however, with the consistency enforcement method, mass conservation is inherent even with low-order quadrature and renders rigorous remap schemes such as CSLAM(which was previously limited to gnomonic cubed-sphere grids) mass conservative on any spherical grid.