Page 141 - Introduction to Computational Fluid Dynamics

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5.3 Method of Solution 2D CONVECTION – CARTESIAN GRIDS
5.3.1 Iterative Solvers
Equations 5.12 for any and Equation 5.60 for p have the same form, which for

any node (i, j) can be generalised as
(AP i, j + Sp i, j ) l+1 = AE i, j l+1 + AW i, j l+1
i, j i+1, j i−1, j
+ AN i, j l+1 + AS i, j l+1 + Su i, j , (5.65)
i, j+1 i, j−1
o
where Su = D, AP = AE + AW + AN + AS, and Sp = (ρ V / t). Note
that Su and Sp can be further augmented to effect underrelaxation, boundary
conditions, and to some extent domain complexity. If there are IN nodes in
the i direction and JN nodes in the j direction, Equation 5.65 represents a set
of (IN − 2) × (JN − 2) equations for the interior nodes for each . These
equations can be solved by matrix-inversion-type direct methods. However, in
multidimensional convection, iterative methods are usually preferred in which
Equation 5.65 is solved sequentially for each . There are two extensively used
methods of this type: GS and alternating direction integration (ADI).

Gauss–Seidel (GS) Method
In the GS method, for each , coefficients AE, AW, AN, AS, Su, and Sp
are evaluated based on values at iteration level l for each node (i, j), i = 2
to IN − 1 and j = 2, JN − 1. Then the nodal value is updated in a double DO loop:

DO1J=2, JN-1
DO1I=2, IN-1
ANUM = AE (I, J)*FI(I+1, J) + AW(I, J)*FI(I - 1, J)
+ AN(I, J)*FI(I,J+1)+ AS(I, J)*FI(I,J-1)
+ SU(I, J)
ADEN = AP(I, J) + SP(I, J)
FI(I, J) = ANUM / ADEN
1 CONTINUE

This method is sometimes called a point-by-point method because each node
(i, j) is visited in turn. Note that as one progresses from i = 2 and j = 2, some
of the neighbouring values are already updated whereas others still retain their
values at iteration level l. Thus, the net evaluation is really a mixed evaluation. Yet,
at the end of the DO loop, values at all nodes are treated as having (l + 1)-level
values. Convergence is declared when the residuals (see the next subsection) fall
below a certain low value. This iterative method, though very robust and simple to
implement, is very slow to converge.