Conclusion
We compared 3 algorithms based on their ability to handle activity
correction in equilibrium chemistry solvers.
The full Newton algorithm is the most integrated algorithm from a
mathematical point of view. Nevertheless, we found it to be the slowest
and weakest algorithm. We suppose this algorithm increases the
nonlinearity of the chemical system by injecting activity corrections
into the mass action equations and conservation laws. It increases the
condition number of the Jacobian matrix, as shown by comparing and . It
has been shown [14, 21] that a condition number that is too high
leads to inaccurate steps in the Newton methods, leading to numerical
difficulties or nonconvergence. Because chemical equilibrium computation
is still a highly nonlinear problem, increasing its nonlinearity by
injecting activity correction seems to be an inefficient choice.
The inner fixed-point algorithm includes an intermediate integration of
activity correction into the Newton loop. Both loops, Newton for the
mass action equations and conservation laws and fixed-point for activity
correction, run together. In this way, changes induced by activity
correction disturb the Newton minimization. This point explains the
convergence difficulties of the inner fixed-point algorithm when
activity correction becomes important.
The outer fixed-point algorithm proposes a complete separation between
the Newton and activity correction loops. In this way, nonlinearity
induced by activity correction cannot disturb the Newton convergence,
and the condition number of the Jacobian matrix is lower than that
obtained by the full Newton algorithm. This leads to a more stable and
robust algorithm. We found that the outer fixed-point algorithm is the
fastest in terms of CPU times for one Newton iteration, usually faster
than or equivalent to the other algorithms in terms of the number of
required Newton iterations and the most robust.
According to the results presented here, we recommend the outer
fixed-point algorithm. This algorithm is the least time consuming for
one Newton iteration, it usually requires the fewest number of
iterations, and it is the most robust and least sensitive to the initial
guess. Moreover, its implementation with existing codes is very simple
and requires very few modifications.
Acknowledgements
The authors acknowledge the French programme NEEDS for its
financial support to the project NewSolChem
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