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Show and correction procedure wi1] dominate the iterative process, often using 5 or more successively coarser grids to approximate the errors for the solution on the finest grid. This multilevel process is il1ustrated in Figure 7 (17]. At each grid level, a certain amount of computational work must be performed. This trork in a unigrid context has been known as iteration . In the past, it was hoped that each iteration would provide maximum solving efficiency. The multigrid algorithn1 itself is a far more efficient solver than any single iteration. Rather, at these steps the multigrid algorithm requires an efficient smoother. That is it is necessary to smooth the solution so that it can be easily approximated on a coarser grid. The design of this smoothing process must be done in the context of the complete system of equations that are being solved. In this respect, the SIMPLE based methods [19] which have been used previously in many fluid dynarrUc codes to couple the momentum and continuity equations are insufficient. Instead, a concept called Distributive REla:ration is used [20]. However, its' advantages can be sununarized as follows: 1) it has a more fundamentally sound mathematical derivation; 2) when used properly \\'ith multigrid methods, no under-relaxation is required; and 3) for complex systems of equations (such as coal combustion), it can be used to more efficiently couple not only the momentum and continuity equations, but also other transport equations, such as those which describe turbulence and rrUxing . In addition to establishing a mechanism by which converged solutions can be rapidly obtained, multigrid methods also provide several other advantages. One of these is that not all of the grid levels necessarily need to encompass the entire computational domain. For instance, if only the burner region of a furnace needs to have higher accurac) (finer grid resolution) , then there can be a set of grid levels, with varying grid resolution which fill the entire geometry, and one grid level with even finer resolution which encompasses only this local inlet region. In this way, computational time is not wasted trying to calculate more accurate predictions in regions where this accuracy is not needed. This concept is called local grid refinEmEnt [21]. 5.2 Results The implementation of multigrid methods requires a strongly systematic approach. These methods are very sensitive, and their performance will degrade significantly if each aspect of the code is not properly addressed. Very simple equations are developed first, and then both ph) sical and numerical complexities are added one at a time. After each step in the development procedure, care is taken to insure that the performance renlains within the theoretical limits . Initial development work at this La boratory centered on axisynlmetric geon1etries . 14 |
