Orthogonal Grid Generation,
COLUMBIA UNIV NEW YORK DEPT OF APPLIED PHYSICS AND NUCLEAR ENGINEERING
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Coordinate transformations have been a key element in most numerical solutions to partial differential equations when either the solution or the region exhibits some geometric complexity. The grids from the transformations are chosen to represent the region boundaries with coordinate curves or surfaces and to adequately resolve the solution by clustering points, curves, or surfaces. The coordinates can be used separately or in a composite fashion to appropriately discretize a configuration with a topological complication in addition to that of basic geometry. In either case, the discretization is well structured since the space of computational variables is rectilinear. Solution procedures developed in a Cartesian setting can then be applied along with the corresponding simplicity in the organization of computational data. To gain the clear advantages of a well-structured discretization, the original partial differential equations must be expressed relative to the grid. The result is usually an increase in the complexity of the equations. This increase is greatest with nonorthogonal coordinates, is fairly mild with orthogonality and is the least with conformal transformations. To choose between the various types of coordinates, it must be considered which constraints are needed for a given problem.