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The root mean square (RMS) residual will be used to estimate the converging slope as
μ
( m)
= R
( m)
/ R
( m−1)
(m) is an iteration counter.
. The weighting reaction R ig is fixed to absorption Σ a ϕ ig . The superscript
Step 5. Modify the over-relaxation factor ω. In the first L e iterations, the initial value of ω 0 will be
used. At each iteration, the value of ω e =1/(1-μ (m) ) is tested. If all the values of ω e in the last
L e iterations agree within the given extrapolation criterion ε e , an extrapolation takes place
using the most recent value of ω e . The testing for a possible extrapolation is suppressed
during the L d iterations following the extrapolation. The value λ (the estimate of the
eigenvalue of the matrix considered) is computed as λ=(μ (m) -1+ω 0 ) and a new ω is obtained
as ω 1 = 2/(2-λ) to be used after the next iteration.
If an increase of the residual is detected during the iteration, a moderate value of ω is
selected as
ω = ( ω × f
2 1 under
)
1/ 2
.
Step 6.
Obtain the new fluxes by the over-relaxation;
ϕ
( m+
2) ( )
( 1/ / − ) .
( m+
1) ( m)
m
ig = ϕig
+ ω ϕig
C ϕig
The loop from Step 3 to Step 6 is repeated until the residual R (m) is less than ε e or the iteration
counter m exceeds L in . The quantities ε, ω 0 , ε e , f under , L e , L d , and L in are input numbers.
In the practical use of this procedure, we find that the scaling by C is effective to accelerate the
convergence, but, we encounter some difficulties in attaining the convergence. One problem occurs in
a weakly absorbing case where a slow convergence rate is observed through the iteration. Once an
extrapolation is taken place, while it greatly reduces the RMS residual, the new over-relaxation factor
ω 1 which takes the value close to 2.0, say, 1.8 after the extrapolation, causes growth of the residual in
most cases. The following procedure which is activated when the increase of the residual is detected in
order to have a moderate over-relaxation factor ω 2 helps to escape from such a catastrophe. Another
problem happens in a strongly absorbing case where we encounter also growth of the residual. It is the
case in which we can expect a rapid convergence. It is thought that because the secondary eigenvalue
of the matrix, λ is not far from the largest eigenvalue λ 0 , the spectral radius of the higher mode in the
modified matrix might exceed unity. We can escape from this trouble by feeding a relatively low f under ,
say 0.5, which suppresses the new factor ω 2 below unity. In other word, in a strongly absorbing case,
an under-relaxation is required.
As the computer time required for the iterative process is much shorter than that for the
preparation of collision probabilities, the optimum use of the above procedure is not essential. We may
suppress the extrapolation by feeding the strict criterion ε e not to activate the extrapolation. Against the
case where the divergence may occur, the low value of f under can prevent the divergence.
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