Abstract. We describe a logically rectangular quadrilateral grid for solving PDEs on the sphere. The grid is logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement.
In particular, we show that the high-resolution wave-propagation algorithm implemented in Clawpack can be effectively used to approximate hyperbolic problems on this grid. Since the ratio between the largest and smallest grid is around 2 for this grid, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with the latitude-longitude grid.
We show that for scalar advection of a smooth solution, the finite volume schemes available in Clawpack produce results which are second order, and are comparable to the results obtained on a lat-long grid. We also solve the shallow water wave equation and show that one obtains reasonable results. We also show that these calculations can be performed on a hierarchy of adaptively refined grids, using a grid with simple periodic boundary conditions in both directions.
Preprint: sphere.ps.gz... sphere.pdf
Matlab routines and fortran codes: (Still under construction...)
bibtex entry:
@InProceedings{dac-ch-rjl:hyp06,
author = "D. A. Calhoun and C. Helzel and R. J. LeVeque",
title = "A finite volume grid for solving hyperbolic problems on
the sphere",
booktitle = "Hyperbolic Problems: Theory, Numerics, Applications,
Proc. 11'th Intl. Conf. on Hyperbolic Problems",
year = "2006",
pages = "to appear",
}