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

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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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.

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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