Multigrid solvers

In this section the basic concepts of the multigrid solvers are outlined. Details of the advanced multigrid procedures may be found in the appropriate literature [23-25, 79, 91, 92, 188, 189, 196, 249, 256]. 

Multigrid acceleration of the Gauss-Seidel point-iterative method is currently used in many commercial CFD codes to solve the system of algebraic equations resulting from the discretization of the governing equations. For this reason, the basic principles and nomenclature must be known by the users of commercial codes and in particular for researchers that are making their own codes. 

Consider a large system of algebraic equations arising from the discretization of the governing equations on a reactor flow domain  

Several stop criteria can be defined in terms of different norms of the residual [205]. The general p-norm of a vectoris defined as r p = (X =i When p tends to infinity, the vector norm 

If we solve this system with an iterative method, we obtain an intermediate solution x after v iterations. This intermediate solution does not satisfy (12.563) exactly, and to determine the error in x we define the residual as follows  

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In the last decade, most new algorithms, schemes, solvers, and preconditioners have found their way into most commercial software packages. Multigrid solvers are also available. Furthermore, all CFD vendors have developed powerful pre- and post processing routines. 

Wheatcraft et al. (1991) considered flow and solute transport in a medium composed of high and low sat distributed according to a Sierpinski carpet fractal, reminiscent of low permeability pebbles distributed in a high permeability matrix. A multigrid solver was used to compute the flow field (Fig. 3 1B) and a particletracking algorithm was used to determine the tracer motion. No diffusion was considered. They found that dispersion increased with the scale of the simulation faster than could be predicted with other models. 

H. Chen, C.-K. Cheng, N. -C. Chou, A. B. Kahng. An algebraic multigrid solver for analytical placement with layout based clustering. In Proc. Design Automation Conf., 2003, pp. 794 - 799. 

Figure 4 The V-cycle in a multigrid solver. An initial guess is made on the fine level, iterations are performed, and then the approximate solution is passed to the coarser level. This process is repeated until the coarsest grid is reached. The solver then progresses through interpolation (correction) steps hack to the finest level. The V-cycle can be repeated.

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