CONSERVATIVE SHAPE-PRESERVING 2-DIMENSIONAL TRANSPORT ON A SPHERICAL REDUCED GRID

A new discretization of the transport equation for two-dimensional tra nsport is introduced. The scheme is two time level, shape preserving, and solves the transport equation in flux form. It uses an upwind-bias ed stencil of points. To ameliorate the very restrictive constraint on the length of the time step appearing with a regular (equiangular) gr id near the pole (generated by the Courant-Friedrichs-Lewy restriction ), the scheme is generalized to work on a reduced grid. Application on the reduced grid allows a much longer time step to be used. The metho d is applied to the test of advection of a coherent structure by solid body rotation on the sphere over the poles. The scheme is shown to be as accurate as current semi-Lagrangian algorithms and is inherently c onservative. Tests that use operator splitting in its simplest form (w here the 2D transport operator is approximated by applying a sequence of ID operators for a nondivergent flow field) reveal large errors com pared to the proposed unsplit scheme and suggest that the divergence c ompensation term ought to be included in split formulations in this co mputational geometry.