A finite volume grid for solving hyperbolic problems on the sphere
by Donna A. Calhoun, Christiane Helzel, and Randall J. LeVeque,
To appear in Proceedings of the
Eleventh Int'l Conference on Hyperbolic Problems, Lyon, 2006.
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", }