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Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains. (English) Zbl 1155.65061

Summary: We describe a class of logically rectangular quadrilateral and hexahedral grids for solving partial differential equations (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.
A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the high-resolution wave-propagation algorithm implemented in clawpack can be used effectively to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grids 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65Y15 Packaged methods for numerical algorithms
35L60 First-order nonlinear hyperbolic equations
58J45 Hyperbolic equations on manifolds
35-04 Software, source code, etc. for problems pertaining to partial differential equations

Software:

CLAWPACK; Matlab; RKC
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