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

arXiv:1105.2970v1 [physics.comp-ph] 15 May 2011 Comment on “A note on generalized radial mesh generation for plasma electronic structure” Jean-Christophe Pain1 CEA, DAM, DIF, F-91297 Arpajon, France 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[α] using high-order coefficients obtained by analytic differentiation of the implicit definition using sym- bolic manipulation). In the present comment, we present a direct proof of that equation. The log-linear radial mesh is defined by the implicit equation [1]: kh = αrk rc + ln rk rc  . (1) Substracting Eq. (1) for k=1 from Eq. (1) for k=n gives (n −1)h = αrn −r1 rc + ln rn r1  . (2) Since one has h0 ≜ 1 n −1 ln rn r1  (3) and rc ≜ αr1 W[αeh], (4) 1jean-christophe.pain@cea.fr 1 Eq. (2) becomes (n −1)(h −h0) r1 rn −r1 = W[αeh]. (5) Equation (3) can be re-written as rn = r1 e(n−1)h0, (6) which enables one to write Eq. (5) as (h −h0) d eh0 = W[αeh], (7) where W represents Lambert’s function and d ≜enh0 −eh0 n −1 . (8) Equation (7) is equivalent to (h −h0) d eh0 e (h−h0) d eh0 = αeh, (9) i.e. (h0 −h) d e(h0−h)( d−eh0 d ) = −α, (10) which leads, by multiplying both sides by s ≜d −eh0 = enh0−neh0 n−1 , to: (h0 −h)s d e(h0−h) s d = −αs, (11) which is equivalent to h = h0 −d s W[−αs], (12) which is the result of Ref. [1]. References [1] B.G. Wilson and V. Sonnad, High Energy Density Phys. 7, 161 (2011). 2

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