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.