Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains

Abstract

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. 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 non-orthogonal, 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.Key words. circular domain, spherical domain, grid transformations, finite volume methods, hyperbolic PDEs AMS subject classifications. 65M06, 65M50, 65Y15, 35L60, 58J451. Introduction. Uniform Cartesian grids are well suited to solving partial differential equations in rectangular regions. Logically rectangular quadrilateral grids work well for many problems where the domain is a deformed rectangle. For example, in an annulus a rectangular grid in polar r-θ coordinates can be used with periodic boundary conditions in the θ direction.It is not so clear that quadrilateral grids are appropriate for solving problems in smooth domains such as a circle (i.e., a planar disk). Polar coordinates could again be used over a rectangular computational domain, but when polar coordinates are used over the full circle there may be numerical difficulties arising at the center of the circle where all of the radial grid lines meet at a single point in physical space. While it is possible to use finite volume methods in this context, these converging grid lines give rise to cells near the center that are much smaller than cells elsewhere in the domain. For explicit finite volume methods the CFL condition may then require a very small time step everywhere. Similar problems arise when using a latitude-longitude grid on the sphere.In this paper we consider a family of quadrilateral and hexahedral grids for the numerical solut...

Wave propagation software, computational science, and reproducible research

“…These days many algorithms can be written in languages such as matlab in a way that is both easy for the reader to comprehend and also executable, with all details intact. For example, we make heavy use of this in the recent paper [15]. We present various grid mappings that define logically rectangular grids in smooth domains without corners, such as those shown later in Figure 1, and all of the matlab codes needed to describe the various mappings are short enough to fit naturally in the paper.…”

A convergence result for finite volume schemes on Riemannian manifolds

“…In particular strongly differing cell sizes impose hard CFL conditions on the timestep (cf. [3]). The grids proposed in [3] also do not satisfy this condition.…”

“…For the convergence analysis we need grids with the property that the curvature of the faces is uniformly bounded under mesh refinement and such that every point on the manifold lies only in a certain number of convex hulls of elements. In [3] different approaches to construct grids on spheres are treated and we refer to [10,18] for geodesic grids on a sphere, which are an important class of examples where these results hold.…”

4π Models of CMEs and ICMEs (Invited Review)

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