Local basis function representations of thermodynamic surfaces: Water at high pressure and temperature as an example

A flexible numerical framework using local basis function is developed in order to better represent thermodynamic properties of fluids as a function of pressure temperature and composition. A new equation of state for water to 100 GPa and 10,000 K is presented. Conventional equations of state are typically based on complex sets of global basis functions that are unique to each application. The use of series expansions in positive and negative powers of the independent variables (with additional exponential/log/trig. factors in some cases) is common and recent accurate equations of state are assembled from collections of such arbitrary terms. These individually crafted representations do not always fit data within uncertainties, can be difficult to implement, and are not easily modified to account for newer data. Multivariate (tensor) b splines overcome these shortcomings. The underlying basis functions are local, orthonormal and complete. Data can be represented to arbitrary precision and the relationships between model parameters and observational constraints are uncomplicated. The fittings of new data in more extended regimes of the thermodynamic space do not require modifying parameters in previously constrained regions. Models are transportable and easily implemented since robust and efficient standardized routines for evaluation of b splines are available in modern numerical environments. Because a local basis function representation can flexibly fit any surface, the articulation of a priori constraints (associated with expected physical and chemical behavior) is appropriately necessary and explicit during model construction. Basic tools are demonstrated in applications to thermodynamic properties of water where the modified equation of state matches a prior formulation below 1 GPa while providing an excellent fit to measurements at much higher pressures.