Cartesian grid (cut cell) methods for hyperbolic problems
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. Cartesian grid methods
are also called "cut-cell methods" or "embedded boundary methods".
The idea is to use a uniform Cartesian grid over most of the domain with the
Cartesian cells cut into a smaller irregular cell in any cell intersected by
the boundary.
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.
One 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. Developing methods that allow the time
step to be chosen based on the uniform portion of the grid, and that also
achieve good accuracy in the irregular cut cells, is the main focus of this
work.
Some papers:
|
A high-resolution rotated grid method for conservation laws with
embedded geometries
by C. Helzel, M. J. Berger, and R. J. LeVeque,
SIAM J. Sci. Comput. 26 (2005), pp. 785-809.
Info/Download
|
|
H-box methods for the approximation of one-dimensional conservation
laws on irregular grids
by M. J. Berger, C. Helzel and R. J. LeVeque
SIAM J. Numer. Anal., 41 (2003), pp. 893-918.
Info/Download
|
|
Cartesian Grid Methods for Fluid Flow in Complex Geometries,
by R. J. LeVeque and D. Calhoun,
Appears in "Computational Modeling in Biological Fluid Dynamics", (L.
J.
Fauci and S. Gueron, eds.) IMA Volumes in Mathematics and its
Applications
124, pp. 117-143, Springer-Verlag, 2001.
(Proceedings of an
IMA
Workshop
January, 1999.)
Info/Download
|
|
Finite Volume Methods for Irregular One-Dimensional Grids
by M. J. Berger, R. J. LeVeque, and L. G. Stern,
Proc. Symp. Appl. Math., 48 (1994) 255-259.
Info/Download
|
|
A Rotated Difference Scheme for Cartesian Grids in
Complex Geometries
by M. Berger and R. J. LeVeque,
AIAA Paper CP-91-1602, 1991
Info/Download
|
|
Stable boundary conditions for Cartesian grid calculations,
by M. Berger and R. J. LeVeque,
Computing Systems in Engineering, 1 (1990) 305-311.
Info/Download
|
|
Cartesian meshes and adaptive mesh refinement for
hyperbolic partial differential equations
by M. Berger and R. J. LeVeque,
Proc. Third Int'l Conf. Hyperbolic Problems, Uppsala (B. Engquist and B.
Gustafsson, editors), Studentlitteratur, Lund, 1990, pp. 67-73.
Info/Download
|
|
An adaptive Cartesian mesh algorithm for the Euler equations in
arbitrary geometries
by M. Berger and R. LeVeque,
AIAA-89-1930, 1989
Info/Download
|
|
High resolution finite volume methods on arbitrary grids via
wave
propagation
R. J. LeVeque,
J. Comput. Phys., 78 (1988), 36-63.
Info/Downlo
ad
|
|
Cartesian grid methods for flow in irregular regions
R. J. LeVeque,
Num. Meth. Fl. Dyn. III (K. W. Morton and M. J. Baines, eds.),
1988,
pp. 375-382.
|
Info/Download
Some related sites:
Return to Randy LeVeque's research interests