Figure 2 : Evolution of
|| Y || versus the number of Newton
iterations for the test case with only activity correction
Comparing the three algorithms on the test case with only activity
correction , one can see in Figure 2 that:
- The outer fixed-point algorithm requires the fewest Newton iterations
to reach convergence, whereas the inner algorithm requires the most
iterations. The full Newton algorithm requires an intermediate number
of Newton iterations.
- For the tree algorithms, obtaining the solution at a low ionic
strength requires less Newton iteration than at a high ionic strength.
This point is obvious: for this test case, activity corrections are
the only nonlinearity of the problem, and they are less important at a
low ionic strength than at a high ionic strength.
- The outer fixed-point algorithm runs 3 minimization loops for the
situations with both low and high ionic strength. The first loops
converge at 10 (low) and 23 (high) iterations; the second loop
converges at 15 (low) and 29 (high) iterations. The third loop is theconfirmation loop used to check that no changes in the ionic
strength computation occur and then to confirm the convergence of the
algorithm.
Frequency graphs
We plot graphs of the cumulative ratio of the resolutions that converge
within a given number of Newton iterations. According to the graph, the
algorithm that reaches a cumulative frequency of 1 is said to be robust.
The algorithm that reaches a high cumulative frequency for a low number
of Newton iterations is said to be fast.
Test case with only activity
correction
The test case with only activity correction is a simple chemical
test case. It makes sense only for studying the activity correction
algorithms. It is solved by all the algorithms (see Table 5) within 150
Newton iterations (Figure 3). The fastest algorithm is the outer
fixed-point algorithm, regardless of the ionic strength. Moreover, this
algorithm shows a very low sensitivity to the ionic strength by
resolving the low ionic strength case within 24 or 25 iterations and the
high ionic strength case within 21 iterations regardless of the initial
guess. The inner fixed-point and the full Newton algorithms are much
more sensitive to the ionic strength, with significant increases in the
number of iterations required to converge in the case with a high ionic
strength. For this case, we find that the best algorithm is the outer
fixed-point algorithm, and the inner fixed-point algorithm is the worst
according to the number of Newton iterations. Taking the computing time
of one Newton iteration into account (Table 4), we see that the full
Newton algorithm is the slowest and the outer fixed-point algorithm is
the fastest.