Comparison of the multigrid and ICCG methods in solving the diffusion equation
The performance of the multigrid method and the Incomplete Cholesky Conjugate Gradient (ICCG) method are compared in this paper. These methods are used in the context of solving the diffusion equation numerically. Both methods are described and test cases and computational results presented. The method of lines is used to solve the diffusion equation. Central difference formulas are used to approximate the spatial derivatives. The backward Euler difference operator is used to approximate the time derivatives. The multigrid and ICCG methods are used to solve the implicit equations. The multigrid method used a series of grids on the region over which the diffusion equation is to be solved. Each of the grids contains a subset of the points of the previous grid. The finest grid contains all of the points involved in the spatial differencing. A coefficient matrix containing the coupling coefficients is generated for each grid. On the coarsest grid, the resulting system may be solved directly. The solution is used to interpolate solutions at points on the finer grids. Relaxation sweeps are performed to accelerate the convergence at interpolated points. The details of the multigrid algorithm and the special difficulties the diffusion equation poses are presented in the chapter on the multigrid method. The ICCG method is a variation of the generalized conjugate gradient method. An incomplete Cholesky decomposition is used in splitting the coefficient matrix. Both the generalized and ICCG methods are described. The metric used to compare the multigrid and ICCG methods is the work unit. A work unit corresponds to the amount of work done in performing one relaxation sweep. Since the ICCG method does not involve relaxation sweeps, some correlations between the work involved in one ICCG iteration and a relaxation sweep are made. The two methods are compared on four test cases and the results presented.
- Research Organization:
- Lawrence Livermore National Lab., CA (USA)
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 6948807
- Report Number(s):
- UCRL-53553; ON: DE84014409
- Country of Publication:
- United States
- Language:
- English
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