Abstract
In this paper, we give an overview of a set of methods being developed for solving classical PDE’s in irregular geometries, or in the presence of free boundaries. In this approach, the irregular geometry is represented on a rectangular grid by specifying the intersection of each grid cell with the region on one or the other side of the boundary. This leads to a natural conservative discretization of the solution to the PDE on either side of the boundary. This method has been used for a broad range of free boundary problems, including material interfaces in compressible (Noh and Woodward, 1976; Miller and Puckett, 1996) and incompressible (Hirt and Nichols, 1981; Puckett et al., 1997) flows, shocks (Chern and Colella, 1987; Bell, Colella and Welcome, 1991), and combustion fronts (Bourlioux and Majda, 1995; Hilditch and Colella, 1995; Pilliod and Puckett, 1997). In the case where the surface being represented in this way is an irregular domain boundary, these methods are often referred to as Cartesian grid or embedded boundary methods (Purvis and Burkhalter, 1979; Young et al., 1991). There has been considerable progress in the development of methods for generating the underlying grid description for complex three-dimensional geometries (Aftosmis, Berger and Melton, 1998), which makes this approach particularly attractive for complex engineering problems. These methods have been used in a variety of unsteady fluid dynamics problems, including inviscid compressible flow in three dimensions (Pember et al., 1995a), incompressible flow in two dimensions (Almgren, Bell, Colella and Marthaler, 1997; Calhoun, 1999) and low-Mach number combustion in an axisymmetric burner (Pember et al., 1995b).
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Colella, P. (2001). Volume-of-Fluid Methods for Partial Differential Equations. In: Toro, E.F. (eds) Godunov Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-0663-8_17
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DOI: https://doi.org/10.1007/978-1-4615-0663-8_17
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