A hybrid augmented compact finite volume method for the Thomas–Fermi equation

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.