Conservative Shape-Preserving Two-Dimensional Transport on a Spherical Reduced Grid

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Abstract
A new discretization of the transport equation for two-dimensional transport is introduced. The scheme is two time level, shape preserving, and solves the transport equation in flux form. It uses an upwind-biased stencil of points. To ameliorate the very restrictive constraint on the length of the time step appearing with a regular (equiangular) grid 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 method 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 conservative. Tests that use operator splitting in its simplest form (where the 2D transport operator is approximated by applying a sequence of 1D operators for a nondivergent flow field) reveal large errors compared to the proposed unsplit scheme and suggest that the divergence compensation term ought to be included in split formulations in this computational geometry.

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