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.
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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.
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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.)
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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.
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A Rotated Difference Scheme for Cartesian Grids in Complex Geometries
by M. Berger and R. J. LeVeque, AIAA Paper CP-91-1602, 1991
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Stable boundary conditions for Cartesian grid calculations,
by M. Berger and R. J. LeVeque, Computing Systems in Engineering, 1 (1990) 305-311.
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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.
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An adaptive Cartesian mesh algorithm for the Euler equations in arbitrary geometries
by M. Berger and R. LeVeque, AIAA-89-1930, 1989
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High resolution finite volume methods on arbitrary grids via wave propagation
R. J. LeVeque, J. Comput. Phys., 78 (1988), 36-63.
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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.
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