Efficient Newton-Raphson and implicit Euler methods for solving the HNC equation

In this paper a new Newton-Raphson (NR) method and a new implicit Euler method are developed for numerically solving the HNC and related equations for simple fluids. The new methods are efficient, stable and easy to implement. The implicit Euler method is derived through a reformulation of the usual HNC problem into the form of an initial value problem (IVP). The IVP formulation allows application of specialized numerical techniques developed for the solution of stiff systems of differential equations. The implicit Euler method is shown to have excellent stability and rate of convergence. The relationship between the implicit Euler method and the NR method is elucidated, and it is shown that in the limit of infinite step-size, the implicit Euler method reduces to the NR method.