The range of ω variation,
ω_{min} ≤ ω < ω_{max}
depends on ω_{0} for the lower limit according to the following
equation:
ω_{min} = ω_{0}  k • (ω_{0}  1) • (ω_{0}
2),
whereby k is a predefined parameter between 1 and 0.
Since k is negative (ω_{min} < ω_{0}),
specifying a greater absolute value of k results in a larger
difference between ω_{min} and ω_{0}, and thus a
larger range. This parameter is set as OMEGA_MIN.
The upper limit of the range of ω variation, ω_{max},
is a set value between ω_{0} and 2. This is defined as OMEGA_MAX.
In the second stage of calculation, the first step is performed with ω
= ω_{min}. The relaxation factor is then increased
incrementally with each iteration until either ω_{max} is
reached or the process begins to diverge, whereupon ω is set back to ω_{min }
and calculation continues.
The increments of ω variation also vary: low values of ω in the
beginning are run through rapidly, but the increment approaches 0
asymptotically as ω approaches the value of 2. The increments are
calculated such that the approximated optimal value of ω_{0} is
reached within a given number of iterations starting from ω_{min}.
The standard number of assumed steps is relatively small number set as
OMEGA_WEIGHT=3.
As already mentioned, ω is set back to ω_{min} as soon
as the iterative results begin to recognisably diverge. The criterium for
discerning whether the solution process is converging or diverging is the
absolute value, Δ_{max}, i.e. the deviation of the results from
one iteration to the next. Hereby a mean value of Δ_{max} is calculated from
the last n steps and compared continually with the current Δ_{max}. The number
of steps included in the mean value for comparison, n, is also defined as a
solverparameter and called OMEGA_TESTNUM. The calculation process is
considered divergent when Δ_{max} equals or exceeds the
comparative mean value. However, it would be impractical to set the
relaxation factor back automatically every time this condition is satisfied
before a certain minimum number of calculation steps has been performed.
This quantity is a further criterium which must be met before an ω
setback occurs. The standard minimum number of iterations here is 23,
defined as OMEGA_VETO. An equation is ultimately considered solved when
the deviation, Δ_{max}, remains smaller than a
defined limit
for a continuous series of a prescribed number of iterations. This quantity
is defined as TERM_NUM. Finally, in order to "smooth" the results of
calculation, a postrun of iterations is performed with a constant
relaxation factor, ω = 1 (defined as OMEGA_POSTRUN=1.0). An increase in
this parameter should be avoided so as not to jeopardise the smoothing
effect of postcalculation. The number of iterations of this stage is
prescribed as POSTRUN=15.
