Development and Verification of Nuclear Calculation Methods for ...

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method for determination of the optimvm relaxation factor, «..
A value of « • o. lower than the anticipated u. is chosen. With this
relaxation factor applied in the original inner iterative problem, the iterative
scheme (4.23), which is the usual power method for the eigenvalue
problem
H(L U ) e=L u e , (4.29)
will determine the largest eigenvalue, the spectral radius f(L ), for
U]
the L matrix.
From
1
the real eigenvalue |i(L u ), where u. * o. , its associated
positive real eigenvalue |i(B) is deduced from equation (4.25). An improved
estimation of the optimum relaxation factor is deduced from equation
(4.27) where u j < « 2 * U D- This relaxation factor will, applied to
the original inner iterative problem, give a new determination of S(L ),
the whole process converging against the optimum relaxation factor ». .
4.2.4. The Extrapolation Method
For improvement of the convergence rate of the outer iterations they
are accelerated by means of an extrapolation procedure. From equations
(4.23) and (4.24) it follows that when a uniform convergence rate is discovered,
it is possible to estimate the true solution for the flux vector
from two successive flux iterates
q> = , + J +1 * *J . (4.30)
j
' - saj
This flux vector is then applied to the fission source for the subsequent
outer iteration.
The relationship between the estimated true flux vector and two flux
iterates as stated in equation (4.30) serves furthermore as a powerful
tool for deciding whether or not a problem has converged. This decision
is very important since a loose criterion will result in a probably false
solution and a firm one in a sometimes too high degree of accuracy, which
consumes too much computer time.
Commonly used criteria relate to successive flux vectors or in more
loose cases successive eigenvalues. The convergence rate is not regarded
in these criteria.
A method which takes the convergence rate into account may be de-