Parametrization and Cartesian representation techniques for robust resolution of chemical equilibria

Chemical equilibria computations, especially those with vanishing species in the aqueous phase, lead to nonlinear systems that are difficult to solve due to gradient blow up. Instead of the commonly used ad hoc treatments, we propose two reformulations of the single-phase chemical equilibrium problem which are in line with the spirit of preconditioning but whose actual aims are to guarantee a better stability of Newton's method. The first reformulation is a parametrization of the graph linking species mole fractions to their chemical potentials. The second is based on an augmented system where this relationship is relaxed for the iterates by means of a Cartesian representation. We theoretically prove the local quadratic convergence of Newton's method for both reformulations. From a numerical point of view, we demonstrate that the two techniques are accurate, allowing to compute equilibria with chemical species having very low concentrations. Moreover, the robustness of our methods combined with a globalization strategy is superior to that of the literature.