|The range of ω variation,
ωmin ≤ ω < ωmax
depends on ω0 for the lower limit according to the following
ωmin = ω0 - k • (ω0 - 1) • (ω0
whereby k is a pre-defined parameter between -1 and 0.
Since k is negative (ωmin < ω0),
specifying a greater absolute value of k results in a larger
difference between ωmin and ω0, and thus a
larger range. This parameter is set as OMEGA_MIN.
The upper limit of the range of ω variation, ωmax,
is a set value between ω0 and 2. This is defined as OMEGA_MAX.
In the second stage of calculation, the first step is performed with ω
= ωmin. The relaxation factor is then increased
incrementally with each iteration until either ωmax is
reached or the process begins to diverge, whereupon ω is set back to ωmin
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 ω0 is
reached within a given number of iterations starting from ωmin.
The standard number of assumed steps is relatively small number set as
As already mentioned, ω is set back to ωmin 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, Δmax, i.e. the deviation of the results from
one iteration to the next. Hereby a mean value of Δmax is calculated from
the last n steps and compared continually with the current Δmax. The number
of steps included in the mean value for comparison, n, is also defined as a
solver-parameter and called OMEGA_TESTNUM.
The calculation process is
considered divergent when Δmax 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
set-back occurs. The standard minimum number of iterations here is 23,
defined as OMEGA_VETO.
An equation is ultimately considered solved when
the deviation, Δmax, remains smaller than a
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 post-run 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 post-calculation. The number of iterations of this stage is
prescribed as POSTRUN=15.