SolveSAPHE: reliable and robust carbonate system pH calculations made in Belgium

SolveSAPHE-v1 (Solver Suite for Alkalinity-PH Equations – version 1 (Munhoven, 2013, DOI:10.5194/gmd-6-1367-2013) was the first to offer carbonate chemistry speciation calculations that were (1) robust, (2) fast, (3) universally convergent for any physically meaningful pair of total alkalinity (AT) and dissolved inorganic carbon (CT) values, and (4) self-starting, i.e., did not require any a priori knowledge of the solution.

The solution approach developed for SolveSAPHE-v1 was recently extended to process AT-CO2, AT-HCO3 and AT-CO3 problems in a similar way (Munhoven, 2021, DOI:10.5194/gmd-14-4225-2021). The mathematical analysis of the modified alkalinity-pH equations reveals that the AT-CO2 and AT-HCO3 problems always have one and only one positive root, for any physically sensible pair of data (i.e., such that, resp., [CO2] > 0 and [HCO3] > 0). For AT-CO3 the situation is completely different: there are pairs of data values for which there is no solution, others for which there is one, and still others for which there are two of them. Similarly to SolveSAPHE-v1, SolveSAPHE-r2 offers automatic root bracketing and efficient initialisation schemes for the iterative solvers. The AT-CO3 problem is furthermore autonomously and completely characterised: for any given pair of data values, the number of solutions is determined and non-overlapping bracketing intervals are calculated.

The numerical solution of the alkalinity-pH equations for the three new pairs is far more difficult than for the AT-CT pair. The AT-CO2 pair is computationally the most demanding. With the Newton-Raphson based solver, it takes about five times as long to solve as the companion AT-CT pair, while AT-CO3 requires about four times as much time. All in all, the secant based solver offers the best performances. It outperforms the Newton-Raphson based one by up to a factor of four and leads to equation residuals that are up to seven orders of magnitude lower. For carbonate speciation problems posed by AT and either one of [CO2], [HCO3] or [CO3] the secant
based routine from SolveSAPHE-r2 is clearly the method of choice; for calculations with AT-CT, the SolveSAPHE-v1 solvers will perform better, due to the mathematically favourable characteristics of the alkalinity-pH equation for that pair.

SolveSAPHE is Free and Open-Source Software, made available on Zenodo.