Optimized random phase approximations for arbitrary reference systems: extremum conditions and thermodynamic consistence

The optimized random phase approximation (ORPA) for classical liquids is re-examined in the framework of the generating functional approach to the integral equations. We show that the two main variants of the approximation correspond to the addition of the same correction to two different first order approximations of the homogeneous liquid free energy. Furthermore, we show that it is possible to consistently use the ORPA with arbitrary reference systems described by continuous potentials and that the same approximation is equivalent to a particular extremum condition for the corresponding generating functional. Finally, it is possible to enforce the thermodynamic consistence between the thermal and the virial route to the equation of state by requiring the global extremum condition on the generating functional.Comment: 8 pages, RevTe

Numerical solution of the unsteady Navier-Stokes equation

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell