The organized discretization of the field has been handicapped by two problems. The first stems from the fact that most fields of interest are arbi­ trarily shaped regions, and thus accurate application of the boundary con­ ditions requires that the discretization conform to the boundaries. This has led to the use of boundary-fitted coordinate systems, as we will see. The second problem in the numerical solution of partial differential equations is the resolution of grid points in regions of the field where very high changes

in the solution occur. These high gradient regions are not known a priori, and therefore presetting of a fine mesh in these regions is out of question. Lack of prior information about these high gradients can render the grid useless since it is not able to resolve these high gradient regions satisfacto­ rily. This makes the numerical simulation itself a waste since important physical phenomena do occur in the high gradient regions. Due to nonlin­ ear phenomena associated with various physical processes such as bound­ ary layer, turbulence, shock formation, and others, there is a tendency for the most important physical processes to occur in high gradient regions. These regions may or may not be associated with solid boundaries, and can also move in time. Thus, the problem of accurate resolution of high gra­ dient regions is important not only from truncation error considerations but also from the physical point of view. The need to have an accurate physical simulation of these high gradient regions, and the lack of a priori information about these regions have led to the development of adaptive grid techniques. This technique automatically controls the grid size depend­ ing upon the driving function-usually the gradient of the solutionmaking the grid very fine in high gradient regions and relatively coarse in low gradient regions.