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= Principles of Grid Generation =

Introduction
A grid is a _________ that covers the physical domain whose objective is to identify the discrete volumes or elements where conservation laws can be applied. Grid generation is the first process involved in computing numerical solutions to the equations that describe a physical process. The result of the solution depends upon the quality of grid. A well-constructed grid can improve the quality of solution whereas, deviations from the numerical solution can be observed with poorly constructed grid. Techniques for creating the cell forms the basis of grid generation. Various methods to generate grids are discussed below.

Differential Equation Methods
Like Algebraic Methods, Differential equation methods are also used to generate grids. The advantage of using the partial differential equations (PDEs) is that the solution of gird generating equations can be exploited to generate the mesh. Grid construction can be done using all three classes of partial differential equations.

Elliptic Schemes
After extensive work done by Crowley (1962) and Winslow (1966) on PDEs by transforming physical domain into computational plane while mapping using Poisson’s equation, Thompson et al. (1974) have worked extensively on elliptic PDEs to generate grids. In Poisson grid generators, the mapping is accomplished by marking the desired grid points (x, y) on the boundary of the physical domain, with the interior point distribution determined through the solution of equations written below

$$ \xi_{xx} + \xi_{yy} = P(\xi, \eta)$$

$$ \eta_{xx} + \eta_{yy} = Q(\xi, \eta)$$

where (ξ, η) represent the co-ordiantes in the computational domain, while P and Q are responsible for point spacing within D. Transforming above equations in computational space yields a set of two elliptical PDEs of the form,

$$\alpha x_{\xi\xi} -2\beta x_{\xi\eta} + \gamma x_{\eta\eta} = -I^2 (Px_{\xi} + Qx_{\eta}) $$

$$\alpha y_{\xi\xi} -2\beta y_{\xi\eta} + \gamma y_{\eta\eta} = -I^2 (Py_{\xi} + Qy_{\eta})$$

where,

$$\alpha = x^2_{\eta} + y^2_{\eta} $$

$$\beta = x_{\eta}x_{\xi} + y_{\xi}y_{\eta} $$

$$\gamma = x^2_{\xi} + y^2_{\xi} $$

$$I = \frac{\delta(x, y)}{\delta(\xi, \eta)} = y_{\eta}x_{\xi} + y_{\xi}x_{\eta} $$

The advantage of using Elliptical PDEs is the solution linked to them is smooth and the resulting grid is smooth. But, specification of P and Q becomes a difficult task adding it to its disadvantages.