On Padé approximants to virial series

Guerrero, André O.; Bassi, Adalberto Bono Maurizio Sacchi

doi:10.1063/1.2958914

Cited by 18 publications

(15 citation statements)

References 19 publications

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“…9, 23, 25-27 B 11 and B 12 , shown here in Table V, also closely match the predictions of Guerrero and Bassi. 28 It has been conjectured 9, 10 that the coefficients of the hard-sphere virial series may become negative at sufficiently large J. To address this interest, we note that, while the predicted virial coefficients extracted from Eq.…”

Section: B Exponential Approximant Hard-sphere Equation Of Statementioning

confidence: 89%

An asymptotically consistent approximant method with application to soft- and hard-sphere fluids

Barlow

Schultz

Weinstein

et al. 2012

The Journal of Chemical Physics

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A modified Padé approximant is used to construct an equation of state, which has the same large-density asymptotic behavior as the model fluid being described, while still retaining the low-density behavior of the virial equation of state (virial series). Within this framework, all sequences of rational functions that are analytic in the physical domain converge to the correct behavior at the same rate, eliminating the ambiguity of choosing the correct form of Padé approximant. The method is applied to fluids composed of "soft" spherical particles with separation distance r interacting through an inverse-power pair potential, φ = ε(σ∕r)(n), where ε and σ are model parameters and n is the "hardness" of the spheres. For n < 9, the approximants provide a significant improvement over the 8-term virial series, when compared against molecular simulation data. For n ≥ 9, both the approximants and the 8-term virial series give an accurate description of the fluid behavior, when compared with simulation data. When taking the limit as n → ∞, an equation of state for hard spheres is obtained, which is closer to simulation data than the 10-term virial series for hard spheres, and is comparable in accuracy to other recently proposed equations of state. By applying a least square fit to the approximants, we obtain a general and accurate soft-sphere equation of state as a function of n, valid over the full range of density in the fluid phase.

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Section: B Exponential Approximant Hard-sphere Equation Of Statementioning

confidence: 89%

An asymptotically consistent approximant method with application to soft- and hard-sphere fluids

Barlow

Schultz

Weinstein

et al. 2012

The Journal of Chemical Physics

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“…8), [4] (chs. 14,19,20), [5], [6], [7], [8], [9], [10]. One of the more well-known techniques is the Padé approximant method [11], which has similarities with the method of asymptotic approximants described herein.…”

Section: Introductionmentioning

confidence: 99%

“…One drawback of the Padé method is that it is not always clear beforehand which sequence is best suited for a given problem [12]. If one has a power series alone, it is difficult to choose a correct Padé form (see [13,14,15] for example). However, if the power series arises from a physical problem, it is likely that some additional conditions or context may be gleaned.…”

Section: Introductionmentioning

confidence: 99%

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Barlow¹,

Stanton²,

Hill³

et al. 2017

Q J Mechanics Appl Math

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A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the 1 method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.

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“…It is worthwhile noting that the choice of the values of coefficients reported on table 1 has been made arbitrarily. Indeed, more recent references could be used, such as [26]. In general, the estimation of a virial coefficient can be obtained from several representations of the same Levin's transformation, as shown in table 2.…”

Section: Methodsmentioning

confidence: 99%

Application of Levin's transformations to virial series

Florindo¹,

Bassi²

2014

Condens. Matter Phys.

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A new method of estimating high-order virial coefficients for fluids composed of equal three-dimensional rigid spheres is proposed. The predicted ½½ and ½¾ values are in good agreement with reliable estimates previously reported. A new application of the Levin's transformations is developed, as well as a new way of using Levin's transformations is suggested. For the virial series of packing factor powers, this method estimates the ½¿ value near 173.

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On Padé approximants to virial series

Cited by 18 publications

References 19 publications

An asymptotically consistent approximant method with application to soft- and hard-sphere fluids

An asymptotically consistent approximant method with application to soft- and hard-sphere fluids

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Application of Levin's transformations to virial series

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