5.18 GAMG method

GAMG is an effective form of multi-grid method used in finite volume CFD. It combines:

  • geometric (“G”) agglomeration to define the structure of the coarse meshes;
  • the algebraic multi-grid (AMG) method, where the matrix eqn is constructed at a coarser level from coefficients at the finer level, rather than by geometric data from the coarse mesh.


Pairwise agglomeration forms coarser meshes by joining pairs of cells at each level of coarsening. In a sweep through the cells, an unpaired cell is paired with the (unpaired) neighbour that shares the face with largest area. This method generally maintains a low aspect ratio (see Sec. 8.2 ) in the resulting agglomerated cells.

The example shows an agglomeration of cell pairs 1-2, 3-4 and 5-6, which form cells 1, 2 and 3 respectively in the coarse mesh. If we solve the equation eqn on a 2D mesh of square cells with eqn, then Eq. (3.2 ) and Eq. (3.5 ) calculate diagonal coefficients eqn and non-zero off-diagonal coefficients eqn.

Algebraic multi-grid creates coefficients in the coarser mesh by summing coefficients eqn and source eqn from the finer mesh. The example produces one row of eqn which is eqn .

If the Laplacian for the coarse mesh were discretised directly, the coefficients in that row would be eqn due to increasing eqn. This discrepancy between agglomerated and calculated coefficients is repeated in subsequent agglomerations.

Multi-grid begins solving/smoothing for eqn (with eqn) at the coarsest mesh. Coarse cell values of eqn are then injected into corresponding cells in the next finest mesh.

With a Laplacian term, the coefficient discrepancy caused by algebraic agglomeration causes eqn to be under-predicted. A correction to eqn can be applied by scaling it by a factor

 r = ---- ---- -----: A \relax \special {t4ht=
This scaling is used when the equation being solved is dominated by a Laplacian term, e.g. a pressure equation, when summing coefficients between agglomerated cells is inaccurate. Scaling is not generally used for transport dominated by advection, since there is no error associated with summing coefficients from the advective derivative.

Equation 5.37 is derived by minimising the error with respect to eqn in the equation eqn. The minimisation is performed by setting eqn for the quadratic form

 2 f( ) = 2- - A r : \relax \special {t4ht=
Notes on CFD: General Principles - 5.18 GAMG method