Consistent thermodynamic derivative estimates for tabular equations of state

Abstract

A valid fluid equation of state (EOS) must satisfy the thermodynamic conditions of consistency (derivation from a free energy) and stability (positive sound speed squared). Numerical simulations of compressible fluid flow for realistic materials require a tabular EOS, but typical software interfaces to such tables based on polynomial or rational interpolants may enforce the stability conditions, but do not enforce the consistency condition and its derivatives. The consistency condition is important for the computation of various dimensionless parameters of an EOS that may involve derivatives of up to second order which are important for the development of more sensitive artificial viscosities and Riemann solvers that accurately model shock structure in regions near phase transitions. We describe a table interface based on the tuned regression method, which is derived from a constrained local least-squares regression technique. It is applied to several SESAME EOS showing how the consistency and stability conditions can be satisfied to round-off while computing first and second derivatives with demonstrated second-order convergence. An improvement of 14 orders of magnitude over conventional derivatives is demonstrated, although the method is apparently two orders of magnitude slower, due to the fact that every evaluation requires solving an 11-dimensional nonlinear system. Application is made to the computation of the fundamental derivative.