Comment on "A note on generalized radial mesh generation for plasma electronic structure"

TL;DR

This paper provides a direct proof for a previously proposed implicit equation involving Lambert's function, which is used for generating analytic log-linear radial meshes in plasma physics.

Contribution

It offers the first direct proof of the implicit equation for parameter h, clarifying its mathematical foundation.

Findings

The implicit equation involving Lambert's W function is proven directly.

The proof confirms the validity of the equation used in radial mesh generation.

This work enhances the theoretical understanding of mesh generation methods.

Abstract

In a recent note [High Energy Density Phys. 7, 161 (2011)], B.G. Wilson and V. Sonnad proposed a very useful closed form expression for the efficient generation of analytic log-linear radial meshes. The central point of the note is an implicit equation for the parameter h, involving Lambert's function W[x]. The authors mention that they are unaware of any direct proof of this equation (they obtained it by re-summing the Taylor expansion of h using high-order coefficients obtained by analytic differentiation of the implicit definition using symbolic manipulation). In the present comment, we present a direct proof of that equation.