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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