A High-Resolution Cartesian Grid Method for the Approximation of Conservation
Laws in Complex Geometries
by Marsha J. Berger, Christiane Helzel and Randall J. LeVeque
Our efforts are directed towards the development of high-resolution
Cartesian grid methods for the approximation of multidimensional systems
of conservation laws in complex irregular geometries. A Cartesian grid
approach is attractive, since away from the boundary it allows the use
of standard high-resolution shock capturing methods that are more difficult
to develop on unstructured (body fitted) grids. Furthermore, embedded boundary
methods allow a more automated grid generation procedure around complex
objects, which is important especially for three-dimensional problems.
The numerical challenge associated with a Cartesian grid embedded boundary
approach is the so-called small cell problem. Near the embedded
boundary the grid cells may be orders of magnitude smaller than regular
Cartesian grid cells. Since standard explicit finite volume methods take
the time step proportional to the size of a grid cell, this would typically
require small time steps near an embedded boundary. We have developed a
numerical method that overcomes the time step restriction, while seeking
an accurate and conservative approximation near the embedded boundary as
well as in the whole domain. Our approach to overcome the small cell problem
is based on the so-called h-box method suggested in earlier work
by Berger and LeVeque. The basic idea behind this method is to approximate
numerical fluxes at the interface of a small cell based on values specified
over regions of length h, where h depends on the size of
a regular grid cell. In order to obtain a method that is stable for a CFL
condition adequate for grid cells away from the boundary, the flux differences
at cut cells should be of the size of the small irregular cell.
In order to gain a better understanding of these methods, we have studied
the construction of high-resolution h-box methods for one-dimensional
conservation laws on irregular grids. We show that our method is:
-
stable, according to the theory of GKS
-
second order accurate for the advection equation on irregular grids
Numerical studies confirmed the same properties for the approximation
of nonlinear systems, e.g. the Euler equations. This work is presented
in:
H-box methods for the approximation of hyperbolic conservation laws
on irregular grids,
by Marsha J. Berger, Christiane Helzel, and Randall J. LeVeque.
to appear in SIAM J. Numer. Anal.: (.ps.gz)
(.pdf)
Recently we have extended these ideas to the two-dimensional case. In order
to obtain the required cancellation property of numerical fluxes, we use
a rotated grid method. We have developed a method that leads to a second
order accurate approximation of boundary cells that may be orders of magnitude
smaller than regular grid cells. Click here for
results showing the performance of our new embedded boundary method.