6.5 Closure

In this chapter, procedures for solution of transport equations on curvilinear and unstructured meshes have been described. By way of a closure, it will be useful to note a few important points.

1. Both procedures require special effort to generate curvilinear or unstructured grids. Some methods for grid generation are introduced in Chapter 8.

2. On curvilinear grids, the familiar (I, J) structure of Cartesian grids remains available. This permits adoption of the fast converging ADI method (as well as some others discussed in Chapter 9) for solution of discretised equations.

3. On unstructured grids, owing to lack of a regular node-addressing structure, a simple point-by-point GS method must be adopted for solution. It is well known that this method is slow to converge, but the convergence rate can be enhanced by adopting fast matrix-inversion techniques such as CG or GMRES. These techniques for sparse matrices become productive when the number of elements is large.

4. It may surprise the reader to note that the unstructured grid procedure is the most general. Since the procedure can handle any polygonal cells (in two dimensions), the Cartesian and curvilinear grids are already included. In the latter cases, however, the advantages of an (I, J) structure must be sacrificed.

5. The procedure for unstructured grids developed in this chapter can be straight-forwardly extended to 3D polyhedral cells (see Figure 6.30). The only difference in three dimensions is that all evaluations with i=1, 2 must now be carried out over i=1, 2, and...