**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

**Matlab routines and fortran codes:**

- grids
- pdes
- circles.tar.gz Download all grids and pdes

**Related links:**

- Slides from
a talk at BIRS, 2008
**Python Tools for Reproducible Research on Hyperbolic Problems**by R. J. LeVeque. To appear in Computing in Science and Engineering (CiSE) special issue on reproducible research. ... Info/Download**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. .... Info/Download**Wave Propagation Software, Computational Science, and Reproducible Research**by R. J. LeVeque, Proceedings of the International Congress of Mathematicians (M Sanz-Sole, J. Soria, J. L. Varona and J. Verdera, eds.) Madrid, August 22-30, 2006, pp. 1227-1254 .... Info/Download- Hexahedral grids in a cylinder

**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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