Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
by Donna A. Calhoun, Christiane Helzel, and Randall J. LeVeque, SIAM Review 50 (2008), 723-752.

Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the three-dimensional ball. The grids are 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 these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude-longitude grids.

Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reaction-diffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.

Publication: SIAM Review 50 (2008), 723-752. SIR000723.pdf

Intro by Ilse Ipsen: SIR000721.pdf

Some figures and animations

Matlab routines and fortran codes:

Related links:

bibtex entry:

@article{cal-hel-rjl:circles,
  author="D. A. Calhoun and C. Helzel and R. J. LeVeque",
  title="Logically rectangular finite volume grids and methods for
        circular and spherical domains",
  journal="SIAM Review",
  year="2008",
  volume=50,
  pages="723-752",
  misc= "{\verb+http://faculty.washington.edu/rjl/pubs/circles+}"
}

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