ABSTRACT
The traditional finite difference methods have computational simplicity when they are applied for the solution of problems involving regular geometries with uniformly distributed grids over the region. However, their major drawbacks include their inability to handle effectively the solution of problems over arbitrarily shaped complex geometries. When the geometry is irregular, difficulty arise from the boundary conditions, because interpolation is needed between the boundaries and the interior points in order to develop finite-difference expressions for nodes next to the boundaries. Such interpolations produce large errors in the vicinity of strong curvatures and sharp discontinuities. Therefore, it is difficult and inaccurate to solve problems with traditional finite-differences over regions having irregular geometries. The use of coordinate transformation and mapping the irregular region into a regular one over the computational domain is not new. Many transformations are available in which the physical and computational coordinates are related with algebraic expressions. But, such transformations are very difficult to construct except for some relatively simple situations; for most multidimensional cases it is impossible to find a transformation. The coordinate transformation technique advanced by Thompson (1977) alleviates such difficulties because the transformation is obtained automatically from the solution of some partial differential equations on the regular computational domain. Therefore, the scheme combines the geometrical flexibility of the finite element method while maintaining the simplicity of the conventional finite-difference technique. In this approach a curvilinear mesh is generated over the physical domain such that one member of each family of curvilinear coordinate lines is coincident with the boundary contour of the physical domain. Therefore, the scheme is also called the boundary fitted coordinate method.
