Multigrid Methods

Modeling > Transport Equation Solution Methodology > Algebraic Multigrid > Multigrid Methods

Multigrid Methods

The primitive iteration methods described above, while relatively simple to implement, exhibit relatively slow convergence characteristics. They tend to be only effective at removing high-frequency (rapidly varying) components of the error. This suggests that some of the work could be done on a coarse grid, since computations on coarse grids are much less costly and the Gauss-Seidel method converges four times faster on a grid half as fine. Multigrid algorithms do this using the following steps:

Agglomerate cells to form coarse grid levels.

Transfer the residual from a fine level to a a coarser level (known as restriction).

Transfer the correction from a coarse level back to a finer level (known as prolongation).

Multigrid algorithms may be divided into two types: geometric and algebraic.

Geometric multigrid makes use of the grid geometry and the discrete equation at the coarse level to arrive at the linear system to be solved on that level.

Algebraic multigrid derives a coarse level system without reference to the underlying grid geometry or discrete equations. The coarse-grid equations are derived from arithmetic combinations of the fine-grid coefficients.

Since it may not always be straightforward to obtain suitable discrete equations on the coarse levels, algebraic multigrid (AMG) is clearly at an advantage. Therefore, it is used for the solution of all linear systems in STAR-CCM+.

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