PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY

PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FORRELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY

Shang-Hsi Tsai1,2,4

and Jaw-Yen Yang1,3,5

1 Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan; [email protected] Institute of Applied Mechanics, National Taiwan University, Taipei 10617, Taiwan

3 Center for Advanced Study in Theoretical Science (CASTS), National Taiwan University, Taipei 10617, TaiwanReceived 2015 March 24; accepted 2015 June 16; published 2015 July 16

ABSTRACT

A two-point Padé approximant (TPPA) algorithm is developed for the equation of state (EOS) for relativistichydrodynamic systems, which are described by the classical Maxwell–Boltzmann statistics and the semiclassicalFermi–Dirac statistics with complete degeneracy. The underlying rational function is determined by the ratios ofthe macroscopic state variables with various orders of accuracy taken at the extreme relativistic limits. Thenonunique TPPAs are validated by Taubʼs inequality for the consistency of the kinetic theory and the specialtheory of relativity. The proposed TPPA is utilized in deriving the EOS of the dilute gas and in calculating thespecific heat capacity, the adiabatic index function, and the isentropic sound speed of the ideal gas. Some generalguidelines are provided for the application of an arbitrary accuracy requirement. The superiority of the proposedTPPA is manifested in manipulating the constituent polynomials of the approximants, which avoids the arithmeticcomplexity of struggling with the modified Bessel functions and the hyperbolic trigonometric functions arisingfrom the relativistic kinetic theory.

Key words: equation of state – hydrodynamics – methods: miscellaneous

1. INTRODUCTION

The equation of state (EOS) provides the mathematicalrelation for the macroscopic state variables of a hydrodynamicsystem. From the principles of relativistic kinetic theory, themacroscopic state variables of the hydrodynamic system can beobtained by integrating the first few moments of the phasespace distribution function, from which the state variables andthe resultant EOS can be shown to be intrinsically dependenton two primitive variables (deGroot et al. 1980; Cercignani &Kremer 2002; Hakim 2011; Rezzolla & Zanotti 2013):

m c

k Ta

k T, , (1)0

2

B

E

Bz

m= =

in which m0 and Em are the rest mass and the chemical potentialper particle; T is the absolute temperature, and kB and c are theBoltzmann constant and the light speed, respectively. Never-theless, such bivariate dependency for a hydrodynamic systemcan be simplified under certain situations. For instance, bypresenting the ideal gas described by the Maxwell–Boltzmann(MB) statistics in terms of the ratios of the macroscopic statevariables, the fugacity being defined as aexp ( ) is ignorable,and the resultant hydrodynamic system involves the univariatemodified Bessel functions only. Mignone et al. (2005)proposed a univariate EOS for the relativistic MB flow, whichwas first used by Mathews (1971), that has the correct valuesfor the extreme relativistic limit (ERL), namely, the ultra-relativistic limit (URL) and the nonrelativistic limit (NRL),and is consistent with Taubʼs inequality (Taub 1948) for alltemperatures. Ryu et al. (2006) proposed an algebraic adiabaticindex function in terms of the reciprocal of ζ defined in

Equation (1) for the MB flow for the numerical relativistichydrodynamic computation.For the hydrodynamic system described by the Fermi–Dirac

(FD) statistics with complete degeneracy (CD), the Chandra-sekhar–Sommerfeld asymptotic expansion simplifies the under-lying bivariate integrals by the Sommerfeld lemma(Chandrasekhar 1967), and the resultant macroscopic statevariables can be presented as univariate functions of the Fermienergy (Chandrasekhar 1967; Cercignani & Kremer 2002;Rezzolla & Zanotti 2013). Tooper (1969) represented the FDintegrals as the contour integrals by the Melling transform,which gave the series of modified Bessel functions and theChandrasekhar–Sommerfeld asymptotic expansions at thelimits of nondegeneracy and extreme degeneracy, respectively.Eggleton et al. (1973) used a bivariate polynomial with thecoefficients determined by the least squares fit to the exactvalues to approximate the designated integrals over the entirerange of arguments. Bludman & van Riper (1977) exploitedthe recursive relations and developed the polynomials ofdegrees up to seven with the tabulated coefficients to calculatethe rate of electron capture on free nucleons and the neutrinoenergy loss. Aparicio (1998) improved the accuracy of theunderlying FD integral by separating the semiinfinite integra-tion interval into four optimized subintervals and calculated thedesignated integrals by the Gauss–Legendre and the Gauss–Laguerre quadratures, respectively. Mamedov (2012) pre-sented an algorithm based on the binomial expansion theoremsand incomplete gamma functions to calculate the thermo-dynamic functions of the relativistic fermions and bosons withdegeneracy.The Padé approximant has been applied in the past decades

to hasten the solution of the mathematical physics problem(Baker & Gammel 1961; Gibson 1981; Adams 1988;Guillaume & Huard 2000; Telega et al. 2000; Cuyt et al.2006; Banks & Torres 2013). It is particularly favorable for theseries outside the disk of convergence, where the Taylor

The Astrophysical Journal Supplement Series, 219:11 (14pp), 2015 July doi:10.1088/0067-0049/219/1/11© 2015. The American Astronomical Society. All rights reserved.

4 Postdoctoral Fellow.5 Distinguished Professor.

1

expansion fails to converge (McCabe & Murphy 1976; Bender& Orszag 1978; Apresyan 1979; Gonzhlez-Vera & Paiz 1993;Guillaume & Huard 2000). The feasibility of the Padéapproximant for a meromorphic function is accomplished bythe Montessus de Ballore theorem, by which the rationalapproximant converges uniformly in a compact region,excluding the singularities therein (de Montessus de Ballore1905). In the present work, the univariate ratios of themacroscopic state variables provide the distinctive mathema-tical relations as the EOS for the hydrodynamic system, whichcan be approximated by the two-point Padé approximant(TPPA) with the designated order of accuracy. By manipulat-ing the rational function being determined from the ERLs, theTPPA avoids the arithmetic complicatedness of manipulatingthe modified Bessel functions and the hyperbolic trigonometricfunctions, which result from the relativistic kinetic theory forthe hydrodynamic MB and CD systems, respectively.

The present work is organized as follows. Section 2 providesbrief reviews on the relativistic kinetic theory and the elementsof the Padé approximant for the subsequent development. InSection 3, the TPPAs of various orders of accuracy for theratios of the macroscopic state variables are determined fromthe ERLs and are verified by Taubʼs inequality. The proposedTPPAs are validated in Section 4 in deriving the EOS for thedilute gas and in calculating the heat capacity at constantvolume and the adiabatic index function for the ideal flows.The approximants are also applied in calculating the isentropicsound speeds of the MB and CD flows. Section 5 providessome general guidelines for the applications of the arbitraryaccuracy requirement. Series expansions at ERLs and theanalytic derivation for the isentropic sound speed are given inthe appendix.

2. PRELIMINARIES

2.1. Review on Relativistic Kinetic Theory

In the absence of external forces and by ignoring the particlecharges and the mutual-particle interactions, the macroscopicstate variables of a hydrodynamic system at equilibrium can bepresented by the kinetic theory as (deGroot et al. 1980;Cercignani & Kremer 2002; Hakim 2011; Rezzolla & Zanotti2013)

c C J a, (2 )20 21

( )r =

e C J b, (2 )0 22( )=

pC

J c3

; (2 )040( )=

where ρ, e, and p are the rest mass density, the total energydensity, and the pressure; C0 is a constant involving the state of

the particle spins; Jmn( ) is an integration function of the primitive

variables ζ and a:

Ja

dsinh cosh

exp( cosh ). (3)mn

m n( )

0òq qz q

qº- + -

¥

The parameter ϵ indicates the statistics describing the particles:

{ 0: classical Maxwell Boltzmann statistics1: semi classical Fermi Dirac statistics.

(4)= -- -‐

For the MB flow, e p( , , )r calculated with 0= can bepresented as the products of the fugacity with the second-kind

modified Bessel functions K ( )2 z and K ( )3 z (deGroot et al.1980; Cercignani & Kremer 2002; Hakim 2011; Rezzolla &Zanotti 2013):

c C aK

aexp ( )( )

, (5 )20

2rz

z= ´

e C aK K

bexp ( )( ) ( )

, (5 )03 2

2

z z zz

= ´-

p C aK

cexp ( )( )

, (5 )02

2

zz

= ´

in which K ( )2 z and K ( )3 z are calculated explicitly by theformula (Abramowitz & Stegun 1972; Press & Vertter-ling 1992; Gil et al. 2007; Olver et al. 2010)

K I

n k

k

k n k

k n k

( ) ( 1) ln2

( )

1

2( 1)

( )

( 1) 2

( 1)

2

( 1) ( 1)

( 1) ( 1) 2,

(6)

nn

n

k

nk

k n

n

k

k n

1

0

1 2

0

2

å

å

zz

z

z

y y z

= -æèççç

öø÷÷÷

+ -G -G +

æèççç

öø÷÷÷

+- + + + +

G + G + +æèççç

öø÷÷÷

+

=

- -

=

¥ +

where n is the nonnegative integer and

xd

dxx( ) ln ( );y = G

I ( )n z and x( )G are the first-kind modified Bessel function andthe gamma function:

Ik n k

( ) 2( 1) ( 1)

,nk

n k

0

2

åz

z

=

æèççç

öø÷÷÷

G + G + +=

¥

+

x t t dt( ) exp( ) .x

0

1òG = -¥

-

Defining the ratios of the state variables e p( , , )r as

Xc

pY

c

eZ

p

e, , , (7)

2 2r rº º º

the fugacity and C0 are eliminated, and the bivariate MB flowturns into univariate, from which XMB is an exact expression,though YMB and ZMB are univariate functions in terms of K ( )2 zand K ( )3 z , respectively:

X a, (8 )MB z=

YK

K KY b

( )

( ) ( )( ), (8 )MB

2

3 2MB

z zz z z

z=-

º

ZK

K KZ c

( )

( ) ( )( ). (8 )MB

2

3 2MB

zz z z

z=-

º

For the FD system, Jmn( ) with 1= - is in essence the puzzling

bivariate FD integral, which can be simplified by theChandrasekhar–Sommerfeld asymptotic expansion for the CDflow, and the resultant e p( , , )r can be presented as univariatefunctions of the Fermi energy EF (Chandrasekhar 1967;

2

The Astrophysical Journal Supplement Series, 219:11 (14pp), 2015 July Tsai & Yang

Cercignani & Kremer 2002; Rezzolla & Zanotti 2013):

( )cC

a3

1 , (9 )2 0 2 3 2r J= -

( )eC

b32

sinh 4 cosh 4 cosh , (9 )0 1 1J J= éëê

- ùûú

- -

( )( )

pC

c

9612 cosh sinh 4 cosh

8 sinh 2 cosh , (9 )

0 1 1

1

J J

J

= éëê

+

- ùûú

- -

-

in which

E

m c1 , .F

02

J x xº + º

The additional relation is dispensable for the CD flow becauseeach of e p( , , )r is univariate and can be used as the primitivevariable to calculate the other ones by solving ξ from thecorresponding relations given by Equation (9).

Remark. As manifested from the series expansions (42d) and(44d) given in Appendix A, the functions R ( )1 z and R ( )2 x atNRL are removed by the ratios of e p( , , )r of the MB and CDflows. The ERLs for the resultant X Y Z( , , ) are (Chandrasekhar1967; Cercignani & Kremer 2002; Rezzolla & Zanotti 2013)

X Y Z a0, 0,1

3, at URL; (10 )= = =

X Y Z b, 1, 0, at NRL. (10 )= ¥ = =

Henceforth, X Y Z( , , ) are called the abstract state variables(ASVs) to distinguish them from the state variables e p( , , )rfor the hydrodynamic system. Shown in Figure 1 are the ASVsX Y( , ) and X Z( , ) for the MB and CD flows and thecorresponding ERLs (10), respectively. The ASVs areunivariate and thus can be used in presenting the functional

relations defined by the state variables e p( , , )r , including theEOS of the hydrodynamic system. For instance, the enthalpyfunction h and the adiabatic index function γ that defines theratio of the enthalpy to the internal energy for the ideal gas canbe alternatively presented as

he p

c X Y

1 1, (11)

2r=

+= +

p

e cZ

Y

1,

11, (12a)

2g

r=

-+

=-

+

Y X

Yb

11. (12 )=

-+

2.2. Elements of the Padé Approximant

The series expansion for the meromorphic f x( ) with poles atthe origin and infinity can be expressed as (McCabe & Murphy1976; Bender & Orszag 1978)

( )f x c x c x O x x a( ) , 0 (13 )j

N

jj

j

M

jj M

1

0

0

0 10 0

0å å» + + =

--

=

+ +

( )c x c x O x x b, (13 )j

N

jj

j

M

jj N

0 1

1å å» + + +¥=

-¥ -

=

¥ - -¥ ¥

¥

where c j0 and c j

¥ are the expansion coefficients, respectively.The TPPA for f x( ) can be presented in general as a rationalfunction defined by the polynomials P x( )M and Q x( )N :

f xP x

Q xf x( )

( )

( )( ) (14)M

N

M N( , )» º

Figure 1. ASVs for the relativistic MB and CD flows, in which X Y 0= = and Z1

3= at URL, and X = ¥, Y = 1 and Z = 0 at NRL, respectively.

3


where

P x p p x p x

Q x q x q x

( ) ,

( ) 1 .

M MM

N NN

0 1

1

= + + +

= + + +

By the series expansions in Equation (13), pk and qk are solvedby equating the coefficients for terms of the same order in thefollowing equation:

( )Q x f x P x O x x a( ) ( ) ( ) , , (15 )N MM N

01 10- = +

¥+¥

x x x x blim , and lim . (15 )x x

00

1º º

¥+¥

-+

The integers M, M0, N, and N¥ satisfy

M N M N c1, (15 )0+ = + +¥

which is deduced from the orders of accuracy of the seriesexpansions (13) taken at the origin and infinity. The superiorityin using the Padé approximant for the EOS instead of the EOSitself for the relativistic hydrodynamic calculation is that eitherthe analytical tasks or the computational costs in manipulatingthe rational functions are far more convenient to access thanthose for the modified Bessel functions and the hyperbolictrigonometric functions arising from the relativistic kinetictheory. Details will be given in subsequent sections.

3. PADÉ APPROXIMANTS FOR RELATIVISTICHYDRODYNAMIC FLOW

3.1. TPPA for X Y Z( , , )MB

The TPPA for XMB can formally be presented fromEquation (8a) as

X X( ) ( ). (16)MB MB(1,0)z zº

Utilizing the ER expansions (39b) and (43b) associated withthe relations (46) and (47), the TPPA for YMB can be assumedwith M = N as

Yp p q

q q( ) 3

1. (17)N N

NN

NN

NNMB

( , )2

21

1

1

z

zz z z

z z=

+ + + +

+ + +

--

The coefficients ( )p q,i j are solved by the ER expansions (39b)and (43b) from the relations provided by Equation (15) withthe order of accuracy

( )( )

YO

O( ):

at URL

at NRL, (18)N N

M

N MMB( , ) 0

1

2

0

0z

z

z

ì

íïïï

îïïï

+

¥-

where 0z and z¥ are defined by Equation (15b). This relation

reveals that Y ( )N NMB( , ) z can be developed with various orders of

accuracies at ERL subject to the restriction (15c). For instance,let N = 3:

( )( )Y

O

a

( )

35 44 16

105 131 68 16with , ,

(19 )

MB(3,3)

2

2 3 02 5

z

z z z

z z zz z=

+ +

+ + + ¥

( ) ( )O b15 20 8

45 60 32 8with , , (19 )

2

2 3 03 4

z z z

z z zz z=

+ +

+ + + ¥

( ) ( )O c6 9 4

18 27 15 4with , , (19 )

2

2 3 04 3

z z z

z z zz z=

+ +

+ + + ¥

( )O d2 (1 )

6 12 7 2with , . (19 )

2

2 3 05 2z z

z z zz z=

++ + + ¥

The relative difference Y N NMB( , )D of Y N N

MB( , ) with respect to YMB

defined by

YY

Y1 , (20)N N

N N

MB( , ) MB

( , )

MBD º -

are shown in Figure 2, in which N = 3 and YMB is calculated bythe modified Bessel functions by Equation (8b). Note that therelative difference is defined in the same manner hereafter.On the other hand, because XMB

(1,0) is exact and Y XZ= bydefinition (7), the TPPA for ZMB can be straightforwardlycalculated as

( )ZY

XO( )

( )

( )with , , (21)N N

N NM N

MB( 1, ) MB

( , )

MB(1,0) 0

0zzz

z z=- ¢¥

¢

where

M M N N M1, 2 .0 0 0¢ = + ¢ = - ¢

Remark. The fugacity for the relativistic hydrodynamic MBsystem can be obtained from Equation (5) by solving ζ fromthe proposed TPPA for the corresponding ASVs. Meanwhile,the EOS model proposed by Ryu et al. (2006) for the adiabaticindex function (12) can be shown to be identical to ( )MB

(1,1)g zfrom the ERLs (39d) and (43d) because

( )

4

3

5

3 3

13

12 5

9 3( ),MB

(1,1)MBRyug z

z

zzz

g z=+

æèççç

öø÷÷÷

+=

++

º

from which

( )4

3 9 27, 0

5

3

1 3, .

MB(1,1)

2

2

g zz z

z

z zz

= + - +

= - + + ¥

+

Comparing with the ERLs (39d) and (43d), one can deducethat

( )O( ) ( ) with , .MB(1,1)

MBRyu

02g z g z z z= ¥

The TPPA for ( )MBg z with a higher order of accuracy isgiven in Section 4.3.Remark. For the EOS proposed by Mignone et al. (2005)

X YY

1 1

3

1, (22)=

æèççç -

öø÷÷÷

which was first used by Mathews (1971) and can be obtainedby taking the equals sign for Taubʼs inequality (which will bedescribed in Section 3.3). The ASVYTM and the adiabatic indexfunction TMg can be solved from Equations (11) and (12b) as

4


(the subscript TM is adopted following Mignone et al. 2005)

YX

X

X XX

X XX

3 9 4

2,

3 27, 0 ,

13

2

9

8, ;

TM

2

3

2

=- + +

= - +

= - + + ¥

+

and

X X

X

X XX

XX

3 8 9 4

6,

4

3 9 81, 0 ,

5

3

1

2, ;

TM

2

3

g =- + + +

= + - +

= - + ¥

+

respectively. A comparison with Equations (39b) and (43b),(39d) and (43d), demonstrates that the orders of accuracy for

YTM and TMg can be shown as ( )O ,03 2z z¥ and ( )O ,0

2z z¥ ,

respectively. From Equations (18) and (31) in Section 4.3

( )( )

Y Y O

O

with , ,

with , ;

TM MB(2,2)

03 2

TM MB(1,1)

02

z z

g g z z

»

»

¥

¥

where

Y(3 2 )

9 6 2,

3

2

27, 0 ,

13

2

27

4, ;

MB(2,2)

2

3

3

z zz z

zz z

z zz

=+

+ +

» - +

» - + + ¥

+

and

12 5

9 3,

4

3 9 27, 0 ,

5

3

1, ,

MB(1,1)

2

gzz

z zz

zz

=++

» + - +

» - + ¥

+

respectively.

3.2. TPPA for X Y Z( , , )CD

The TPPA of XCD, YCD, and ZCD can be obtained by theseries expansions at ERLs given in Appendix A. Theunderlying rational functions can be assumed according to

Figure 2. Relative difference distributions for the TPPA Y ( )MB(3,3) z with various orders of accuracy at ERLs, in which X Xlim0

0MB zº =

z +at URL and

X Xlim MB1 1zº =

z¥

¥

- - at NRL.

5


the relations (46), (47), and (48), respectively:

Xp p q

q q

Yp p q

q q

Zp p q

q q

( )

5

2ˆ ˆ 4 ˆ

ˆ ˆ,

( )1 ˆ ˆ

4

3ˆ

1 ˆ ˆ,

( )

2

5ˆ ˆ

1

31 ˆ ˆ

.

N NN

NN

N

NN

N NN

NN

N

NN

N NN

NN

N

NN

CD( 1, )

1 22 1

22

CD( 1, )

1 22 1

1

CD( , )

22

11

1

xx x x

x x x

xx x x

x x

xx x x x

x x

=+ + + +

+ + +

=+ ¢ + + ¢ + ¢

+ ¢ + + ¢

=+ + + +

+ + +

--

- -

--

- -

--

Note that even though Y X ZCD CD CD= , the coefficients ( )p qˆ , ˆ ,

( )p qˆ , ˆ¢ ¢ , and ( )p qˆ , ˆ ¢¢ must be respectively solved by the sameprocedure with the orders of accuracy being estimated by therelation (15c). For instance, let N = 3:

( )Y O

a

( )30 24 4

30 42 21 3with , ,

(23 )

CD(2,3)

2

2 3 02 5x

x xx x x

x x=+ +

+ + + ¥

( )O b70 60 12

70 102 54 9with , , (23 )

2

2 3 03 4x x

x x xx x=

+ ++ + + ¥

( ) ( )O c5 63 56 12

315 469 255 45with , , (23 )

2

2 3 04 3

x x

x x xx x=

+ +

+ + + ¥

( )( )O

d

35 99 90 20

3465 5229 2887 525with , ,

(23 )

2

2 3 05 2

x x

x x xx x=

+ +

+ + + ¥

respectively. Figure 3 demonstrates YCD(2,3)D with various orders

of accuracy, in which

( )

( )

( )

( )

YX

O

XO

XO

XO

4200with ,

2880with ,

1152with ,

512with ,

CD(2,3) 0 5

02

4

03

3

04

2

x

x

x

x

D »

»

»

»

¥

¥

¥

¥

at URL and

( )

( )

( )

( )

YX

O

XO

XO

XO

3125

4756752with ,

125

116424with ,

5

2352with ,

168with ,

CD(2,3)

5

02

4

03

3

04

2

05

x

x

x

x

D »

»

»

»

¥

¥

¥

¥

at NRL, where 0x and x¥ are defined by Equation (15b), and

X X alim 4 at URL, (24 )0 CD xº =x¥

¥

X X blim2

5at NRL, (24 )

0CD

10xº =

x¥

-+

respectively. Similar results can be developed for X ( )CD(2,3) x and

Z ( )CD(3,3) x , respectively.

Figure 3. Relative difference distributions for the TPPA Y ( )CD(2,3) z with various orders of accuracy at ERLs, in which X Xlim 40 CD

1xº =x¥

- at URL and

X Xlim2

50CD

1 xº =x

¥

-+

at NRL, respectively.

6


3.3. Taubʼs Inequality

The macroscopic state variables resolved from the mathe-matically derived TPPA must comply with Taubʼs inequality,which incorporates the consistency for the kinetic theory withthe special theory of relativity (Taub 1948):

e e p c( 3 ) , (25)2 4r- ⩾

from which Mignone et al. (2005) deduced the modeling EOSfor e p( , , )r . The inequality (25) can be recast and expressed interms of the ASVs as

T X Y XY

Y( , )

3

10. (26)

2º -

-⩾

It is suggested from numerical tests without rigorous proof thatthe TPPAs comply with Taubʼs inequality, except for the oneswith lower accuracy. For instance, it is numerically verified thatT X Y( , ) 0MB ⩾ for XMB

(1,0) with Y N NMB( , ) of ( )O ,M N M

02 10 0z z¥

- + ,

where M 10 > , N 5⩽ , and N M2 20> + , except for

( )M N, (1, 2)0 = . The same results can be obtained for

X N NCD( , 1)+ with ( )O ,M N M

02 10 0x x¥

- + and Y N NCD( , 1)+ with

( )O ,M N M0

2 30 0x x¥- + , respectively. The solid and dashed curves

shown in Figure 4 illustrate T X Y( , ) 0⩾ drawn with the exactASVs for the MB and CD gases, respectively. The circlesrepresent the Taubʼs inequalities drawn by the TPPAs of

Y ( )MB(5,5) z with ( )O ,0

5 6z z¥ , and the squares are drawn by

X ( )CD(4,5) x with ( )O ,0

4 5x x¥ and Y ( )CD(4,5) x with ( )O ,0

6 5x x¥ ,respectively.

4. VALIDATION AND APPLICATION

In astrophysics, a gas is typically considered an ideal fluidwhen modeling gaseous flows of low densities and hightemperatures around stars or in interstellar space. The adiabaticindex function is usually considered a constant, for example,γ = 1.4 for the ordinary ideal gas flows and γ = 4/3 for theultrarelativistic flows, respectively. In this section, manipulat-ing the rational functions in various examples demonstrates thesuperiority of the proposed TPPAs, which avoid the arithmeticintricacy of dealing with the modified Bessel functions and thehyperbolic trigonometric functions arising from the kinetictheory.

4.1. EOS for Dilute Gas

The dilute gas is characterized by the vanishing numberdensity nr , which can be calculated by the series expansions(42a) and (44a) with z ¥ and 0x +, respectively:

m c 1.n 02r rº

To the first order of accuracy

Y Z13

2, (27)» -

which requires that the coefficients of the TPPAs satisfy

p q

q

p q

3

2,

ˆ ˆ3

5.

N N

N

1 1

1 1

-=-

¢ - ¢ =-

- -

The coefficients given by Y ( )MB(3,3) z and Y ( )CD

(2,3) x fromEquations (19) and (23) comply with the above conditions.

Figure 4. Taub’s inequality for the MB and CD flows, where TMB and TCD are drawn by the exact ASVs, and TMBPade by XMB

(1,0) and YMB(5,5) with ( )O ,0

5 6z z¥ , and TCDPade by

XCD(4,5) with ( )O ,0

4 5x x¥ and YCD(4,5) with ( )O ,0

6 5x x¥ , respectively.

7


Expressing the relation (27) in terms of the state variablese p( , , )r yields the EOS for the dilute gas that is identical to

that being derived from the principles of the relativistic kinetictheory for the ideal gas (deGroot et al. 1980; Cercignani &Kremer 2002; Rezzolla & Zanotti 2013):

e c p3

2.2r= +

4.2. Specific Heat Capacity

The specific heat capacity for the ideal gas at constantvolume is defined and presented in terms of the ASVs as(Cercignani & Kremer 2002)

cT

ek

Y·

1v

v vB

2

rz

z=

¶¶

æ

èççç

ö

ø÷÷÷÷

= -¶¶

æèççç

öø÷÷÷

according to Equation (1). Substituting YMB and YMB(3,3) from

Equations (8b) and (19c) yieldsc

kG G5 1, (28)

v,MB

B

2 2 2z z z= + - -

( )( )

c

k

3 36 108 123 56 8

6 9 4(29)v,MB

Pade

B

2 3 4

2 2

z z z z

z z=

+ + + +

+ +

where G K K( ) ( )3 2z zº . A comparison of the two formulasdemonstrates the simplicity and convenience of the proposedTPPA because approximating cv,MB

Pade by differentiating YMB(3,3)

only involves manipulating the simple rational functions,whereas one has to deal with the much more complicatedmodified Bessel functions when calculating cv,MB by YMB. Thesolid curve and the circles shown in Figure 5 demonstratec kv,MB B and c kv,MB

PadeB

drawn by Equations (28) and (29), with

the ERLs indicated by the dashed and the dash-dotted lines:

c

k

c

k

c

k

c

k

c

k

c

k

lim lim 3,

lim lim3

2;

v v v

v v v

,UR

B 0

,MB

B 0

,MBPade

B

,NR

B

,MB

B

,MBPade

B

= = =

= = =

z z

z z

¥ ¥

+ +

which are consistent with the values being used in classical gasdynamics. The relative difference cv,MBD can be estimated by

YMB(3,3) given by Equation (19):

( )

( )

( )

( )

c O

O

O

O

263

7350with ,

1

90with ,

1

18with ,

4

3with ,

v,MB 02

02

02

03

03

04

05

05

z z

z z

z z

z z

D »

»

»

»

at URL and

( )( )

( )

( )

c O

O

O

O

175

256with ,

5

16with ,

3

16with ,

1

6with ,

v,MB4 5

3 4

2 3

2

z z

z z

z z

z z

D »

»

»

»

¥ ¥

¥ ¥

¥ ¥

¥ ¥

at NRL, respectively. The specific heat capacity cp,MB and the

corresponding cp,MBPade at constant pressure can be calculated from

Figure 5. Specific heat capacity at constant volume and the ERLs for the MB flow.

8


cv,MB and cv,MBPade because by definition

ch

Tc k .p

pv B=

æèççç¶¶

öø÷÷÷ = +

4.3. Adiabatic Index Function

An ideal gas under a reversible adiabatic process can beshown by the first law of thermodynamics and the relation (34)given in Section 4.4 (Landau & Lifshitz 1980; Huang 1987;Hakim 2011):

dp

d

Z

X

Z

Y

11

(ln )

(ln ), (30)

r=

+ æ

èçççç

-ö

ø

÷÷÷÷÷

c

c

where d d() () cºc and c z= and ξ for the MB and the CDgases, respectively. Substituting the ASVs with the TPPAs inEquation (30) and taking the first terms of the ERLs result inthe adiabatic index in classical gas dynamics for the ideal gas(Hakim 2011). For instance, using the TPPAs obtained inSection 3 with N = 3 yields

dp

d X

p

X

p

4

3or at URL,

5

3or at NRL;

4 3

5 3

r rk

rk

» =

» = ¢

these give the adiabatic indexes 4

3g = and 5

3g = , where κ and

k¢ are constants, respectively.On the other hand, by the definition (12) the adiabatic index

function can be approximated by substituting the TPPA for theASVs or by calculating the TPPA of the adiabatic indexfunction from Equations (39d) and (43d):

p p q

q q( )

4

3

5

31

(31)N NN

NN

N

NNMB

( , )1 1

1

1

g zz z z

z z=

+ + + +

+ + +

--

with the orders of accuracy being determined byEquation (15c). It is intriguing to find that the TPPA for

( )N NMB( , )g z with ( )O ,M N M

02z z¥

¢- ¢ is identical to the approximant

of MBg obtained by substituting into Equation (12) X ( )MB(1,0) z

and Y ( )N NMB( , ) z¢ ¢ with ( )O ,M N M

02z z¢¥

¢- , where M M 1¢ = + andN N 1¢ = + , respectively. Conversely, solving from Equa-tion (31), the approximant for Y ( )MB z is the same as Y ( )N N

MB( , ) z¢ ¢

with ( )O ,M N M0

2z z¢¥

¢- . For instance, the TPPA for ( )MBg z withN = 3 and M = 4

( )O( )24 90 75 20

18 66 51 12with , ,

(32)

MB(3,3)

2 3

2 3 04 3g z

z z zz z z

z z=+ + ++ + + ¥

is identical to that obtained by substituting X ( )MB(1,0) z zº and

( )( )Y

O

( )

2 3 12 12 4

18 72 75 36 8with ,

(33)

MB(4,4)

2 3

2 3 4 05 4

z

z z z z

z z z zz z=

+ + +

+ + + + ¥

into the adiabatic index function (12). Conversely, the resultantapproximant for Y ( )MB z solved from Equation (32) is the same

as that given by Equation (33). The results suggest that( )N N

MB( , )g z can be considered as the EOS for the TPPA of the

ASVs, from which Y ( )N NMB( 1, 1) z+ + can be determined with

improved accuracy. Nevertheless, this relation is invalid for( )N N

CD( , )g x , which has to be calculated from the corresponding

TPPAs of the ASVs or by Equation (31) associated with theERLs (41d) and (45d) for the CD gas, respectively.Figure 6 schematically demonstrates the adiabatic index

functions MBg and CDg , which are indicated by the solid curveand are presented as functions of XMB and XCD, respectively.The approximants MB,CD

Padeg , which are obtained by N NMB,CD( , )g and

by the substitution of the TPPAs of the ASVs, are representedby the circles shown in Figure 6. The dashed and dash-dottedlines indicate the ERLs, which are UR

4

3g = and NR

5

3g = ,

respectively. The relative differences MBgD calculated by MB(3,3)g

and YMB(3,3) with ( )O ,0

4 3z z¥ can be estimated as

( )O

Y

by ,

1

144by ,

MB 04

MB(3,3)

03

MB(3,3)

g z g

z

D »

»

at URL and

Y

3

160by ,

1

40by ,

MB3

MB(3,3)

2MB(3,3)

g z g

z

D »

»

¥

¥

at NRL, respectively. Using the relation (24), the relative

differences CDgD calculated by CD(3,3)g with ( )O ,0

3 4x x¥ and by

the TPPAs X ( )CD(2,3) x , Y ( )CD

(2,3) x , and Z ( )CD(3,3) x with ( )O ,0

2 3x x¥ ,

( )O ,03 4x x¥ , and ( )O ,0

3 4x x¥ , can be estimated as

X

X X Y

X Y Z

17

2592by ,

1

3840by and ,

671

134784by and ,

CD 04

CD(3,3)

02

CD(2,3)

CD(2,3)

04

CD(2,3)

CD(3,3)

g gD »

»

»

at URL and

X

X X Y

X Y Z

25

77616by ,

1

1764by and ,

29

7056by and ,

CD3

CD(3,3)

2CD(2,3)

CD(2,3)

2CD(2,3)

CD(3,3)

g gD »

»

»

¥

¥

¥

at NRL, where X0 and X¥ are given by (24), respectively.

4.4. Isentropic Sound Speed

The isentropic sound speed cs in the equilibrium hydro-dynamic system (Weinberg 1972; Landau & Lifshitz 1980;Huang 1987; Mignone et al. 2005; Hakim 2011) is

cp

e h

p1.

S Ss2

r=

¶¶

=¶¶

9


The subindex S indicates that the entropy of the flow remainsunchanged as the disturbance is traveling. Note that thefollowing thermodynamic relation has been used in deriving cs

(Hakim 2011; Huang 1987; Landau & Lifshitz 1980):

de hd , (34)r=

by which cs2 can be presented in terms of the ASVs as

cZ X Z

Y

(ln ) (ln (1 ))

(ln ). (35)s

2 =+ +c c

c

The series expansions for c( )s2

MB at ERLs are given byEquations (39e) and (43e), from which the TPPA can beassumed according to (46) and (47) as

( )cp p q

q q

1

3

5

31

.

(36)

N N NN

NN

NNs

2

MB

( 1, ) 1 22 1

1

z z z

z z=

+ + + +

+ + +

- -- -

The isentropic sound speed can also be calculated by insertingthe TPPAs of the X, Y, and Z obtained in the previous sectioninto Equation (35). Figure 7 demonstrates c( )s

2MB by

Equation (35) drawn as a solid curve by the exact ASVs; thecircles represent c( )s

2MBASV by substituting XMB

(1,0) and YMB(4,4) with

( )O ,05 4z z¥ into Equation (35) and c( )s

2MB(4,5) by Equation (36)

with ( )O ,05 6z z¥ , respectively. The relative differences can be

presented as

( )c X

X

7039

446400at URL,

53775

51712at NRL,

s2

MB

(4,5)05

5

D »

» ¥

by the dashed curve and

( )c X

X

1

48at URL,

3

40at NRL,

s2

MB

ASV04

3

D »

» ¥

by the dash-dotted curve.The isentropic sound speed for the CD flow calculated from

Equation (35) is an exact rational function, which can formallybe presented as a TPPA of infinite order of accuracy:

( ) ( )c c(2 )

3(1 ). (37)s

2

CD 2 s2

CD

(2,2)x xx

=++

º

The series expansions at ERLs are given by Equations (41e)and (45e). Figure 8 demonstrates c( )s

2CD by Equation (35) as a

solid curve, and the circles represent c( )s2

CDASV by XCD

(3,4), YCD(3,4),

and ZCD(4,4) with ( )O ,0

3 4x x¥ , ( )O ,05 4x x¥ , and ( )O ,0

5 4x x¥ ,respectively. The dashed curve indicates the relative differencewith X0 and X¥ given by (24), respectively:

( )c X

X

3

640at URL,

225

130438at NRL.

s2

CD

ASV03

3

D »

» ¥

Figure 6. Adiabatic index function and the ERLs. Note that MBPadeg and CD

Padeg include the approximants obtained by the TPPA of γ and by the TPPAs of the ASVs,respectively.

10


5. DISCUSSIONS

Because the proposed TPPAs for the EOS are essentially therational functions, which are easy to manipulate even in codingthe computational programs for the roots of the high-orderpolynomials derived from the designated rational functions,some guidelines are supplemented for the applications of anarbitrary accuracy requirement.

The results shown in Figures 2 and 3 indicate that therelative differences of various orders of accuracy are nonuni-form distributions. The convexity of these curves in log scale

manifests the fact that even the EOS approximants of low-orderaccuracy yield satisfactory numerical results as the ASVs XMBand XCD approaching the ERLs, whereas one has to use a largeN to increase the order of accuracy for Y N N

MB( , ) and Y N N

CD( , 1)+ ,

provided that the corresponding XMB and XCD of main concernto the problem are located primarily in the intermediate regionbetween the ERLs. For instance, let N = 4, by which themaximums of the curves YMB

(4,4)D and YCD(3,4)D (not shown) that

occur around X 1MB = and X 1CD = are shifted downwardabout one order with respect to the corresponding ones shownin Figures 2 and 3, respectively. Hence, we conclude that the

Figure 7. Isentropic sound speed and the relative difference distributions for the MB flow calculated by the TPPA of ( )cs2

MB(4,5) and by the TPPAs of YMB

(4,4), respectively.

Figure 8. Isentropic sound speed and the relative difference distributions for the CD flow calculated by the exact ( )cs2

CD(2,3) and by the TPPAs of XCD

(3,4),YCD(3,4), and ZCD

(4,4),respectively.

11


values of XMB and XCD shall be considered when setting up theaccuracy of the approximants Y N N

MB( , ) and Y N N

CD( , 1)+ , respectively.

In general, unless one is restricted by limited computationalcapability or resources, the high-order accuracy approximantsare recommended for all applications.

Besides, the proposed EOS approximants from which onededuces the odd-order polynomial are considered if solving theroots of the deduced polynomial is inevitable in the numericalscheme. This is because there exists at least one real root for thepolynomial of odd order, whereas there may be none for that ofeven order (i.e., all of the roots are complex conjugates). Notethat the physical properties and the restrictions, particularly theones being induced from the relativistic condition, such as theprohibited superluminal motion for the material particles, haveto be checked for each one of the multiple roots if they exist inthe tailor-made numerical schemes.

Setting up the accuracy for the EOS approximants is alsodependent on the local properties of the flow field. In general,low-accuracy EOS approximants are sufficient to furnishsatisfactory numerical results for less critical regions in theflow. On the other hand, the EOS approximants of a higherorder of accuracy are needed to resolve the macroscopicphysical quantities with severe local gradients, where therefined grids are deployed to capture the detailed flowstructures, for instance, the regions near the adjacent flowlayers of distinct properties (Hardee & Stone 1997; Stoneet al. 1997; Shore 2008) and the domains where the shockscollide (Shore 2008) and interact with the interstellar medium(Xu & Stone 1995; Shore 2008).

6. SUMMARY

The feasibility of the present methodology depends on thereduction of the bivariate state variables to the univariateASVs, which provides the mathematical relation as the EOS forthe classical MB and the completely degenerate semiclassicalFD hydrodynamic systems. The nonunique TPPAs of higheraccuracy were numerically verified by Taubʼs inequality, whichvalidated the consistency of the kinetic theory and the specialtheory of relativity for the resultant macroscopic state variables.The superiority of the proposed approximants over the ordinaryEOS was manifested by various examples. Either the analyticaltasks or the computational costs in manipulating the underlyingrational functions were shown to be far more accessible thanthat from the modified Bessel functions and the hyperbolictrigonometric functions emerging from the relativistic kinetictheory.

The present work studied the Padé approximant for the EOSby the relativistic kinetic theory for the ideal flow free fromexternal force fields. Some general guidelines were providedfor the applications of an arbitrary accuracy requirement. Theextension and application of the present methodology to asophisticated hydrodynamic system, such as gas mixtures withmutual interactions, including the annihilation and creation forthe fermions, and the incorporation of electromagnetic andgravitational fields to solve the phase space distributionfunction for the charged particle from the Boltzmann–Vlasovequation, require further investigation.

This research was supported by the National Science Councilunder grant No. NSC 100-2811-E-002-035. The authors aregrateful to Dr. Muljadi (Research Associate, Earth Science andEngineering Department, Imperial College London) and PhD

candidate Manuel A. Diaz (Institute of Applied Mechanics,National Taiwan University, Taiwan) for valuable discussions.

APPENDIX ASERIES EXPANSIONS AT EXTREME

RELATIVISTIC LIMITS

The series expansions of the state variables e p( , , )r and theASVs X Y Z( , , ) are given for determining the TPPA. Note thatthe constant C0 has been ignored.

A.1. Series Expansions at URLs

For the MB flow at URL: 0z +

c O a2

14

( ), (38 )23

2

rz

zz=

æ

èçççç

-ö

ø÷÷÷÷+

e O b6

112

(1), (38 )4

2

zz

=æ

èçççç

-ö

ø÷÷÷÷+

p O c2

14

(1); (38 )4

2

zz

=æ

èçççç

-ö

ø÷÷÷÷+

X a, (39 )MB z=

( )Y O b3

16

, (39 )MB

25z z

z=æ

èçççç

-ö

ø÷÷÷÷+

( )Z O c1

3 18; (39 )MB

24z

z= - +

( )O d4

3 9 54

2

81, (39 )MB

2 34g

z z zz= + - - +

( )( )c O e1

3 36

11

864. (39 )s

2

MB

2 46z z

z= - + +

For the CD flow at URL: x ¥

c a4

31

3 3

2

1

2

3

8, (40 )2 3

2 3 4r

px

x x x x=

æ

èçççç

+ + - + +ö

ø÷÷÷÷

e O b14 5 2

(1), (40 )42 3

pxx x x

=æ

èçççç

+ + +ö

ø÷÷÷÷+

p O c3

14 3 2

(1); (40 )42 3

px

x x x=

æ

èçççç

+ + -ö

ø÷÷÷÷+

X O a4

11 5

2

1, (41 )CD 2 4x x x x

=æ

èçççç

- +ö

ø÷÷÷÷+

æ

èçççç

ö

ø÷÷÷÷

Y O b4

31

1 1

2

1, (41 )CD 2 4x x x x

=æ

èçççç

- +ö

ø÷÷÷÷+

æ

èçççç

ö

ø÷÷÷÷

Z O c1

3

2

3

4

3

1; (41 )CD 2 3 4x x x

= - + +æ

èçççç

ö

ø÷÷÷÷

O d

4

3

4

9

14

27

22

81

1, (41 )

CD 2

3 4

gx x

x x

= + -

+ +æ

èçççç

ö

ø÷÷÷÷

12


( )c e1

3

1

3

2

3

1. (41 )s

2

CD 2 3 4x x x= - + - +

A.2. Series Expansions at NRLs

For the MB flow at NRL: z +¥

c R

a

( ) 115

8

105

128

315

1024

10395

32768, (42 )

21 2

3 4

r zz z

z z

=æ

èçççç

+ +

- + +ö

ø÷÷÷÷

e R

b

( )

127

8

705

128

2625

1024

34965

32768,

(42 )

1

2 3 4

z

z z z z

=æ

èçççç

+ + + - +ö

ø÷÷÷÷

pR

c

( )1

15

8

105

128

315

1024

10395

32768, (42 )

1

2

3 4

zz z z

z z

=æ

èçççç

+ +

- + +ö

ø÷÷÷÷

R d( )2

exp( ); (42 )1 3 2

zp z

zº

-

X a, (43 )MB z=

Y b13

2

3

8

33

8

1377

128, (43 )MB 2 3 4z z z z

= - + + - +

Z c1

13

2

3

8

33

8

1377

128; (43 )MB 2 3 4z z z z z

=æ

èçççç

- + + - +ö

ø÷÷÷÷

d5

3

5

6

15

8

185

48

2735

384, (43 )MB 2 3 4

gz z z z

= - + - + +

( )c

e

5

31

7

2

87

8

269

8

13975

128. (43 )

s2

MB 2

3 4

z z z

z z

=æ

èçççç

- +

- + +ö

ø÷÷÷÷

For the CD flow at NRL: 0x +

c R

a

( ) 13

4

3

32

128

3

2048, (44 )

22

2

3 4

r xx x

x x

=æ

èçççç

+ +

- + +ö

ø÷÷÷÷

e R

b

( ) 127

20

141

224

25

384

111

22528, (44 )

2

2 3

4

xx x x

x

=æ

èçççç

+ + +

- +ö

ø÷÷÷÷

p R

c

2

5( ) 1

15

28

5

96

5

1408

15

26624, (44 )

2

2

3 4

xx

x x

x x

=æ

èçççç

+ +

- + +ö

ø÷÷÷÷

R d( )8 2

3; (44 )2

3 2x pxº

X

a

5

21

3

14

43

588

717

30184

358663

49441392, (45 )

CD

2 3

4

xx x x

x

=æ

èçççç

+ - +

- +ö

ø÷÷÷÷

Y b13

5

48

175

172

2625

13051

336875, (45 )CD

2 3 4x x x x= - + - - +

Z

c

2

51

57

70

274

525

105373

404250

358577

4379375; (45 )

CD

2

3 4

x x x

x x

=æ

èçççç

- +

- + +ö

ø÷÷÷÷

d5

3

5

21

220

1323

11695

101871. (45 )CD

2 3

gx x x

= - + - +

( )c e2

31

3

22

5

23 . (45 )s

2

CD2

34x x

xx

x=æ

èçççç

- + - + +ö

ø÷÷÷÷

APPENDIX BRELATIONS FOR P x( )M AND Q x( )N

Relations that simplify the determination of the coefficientsof P x( )M and Q x( )N for the TPPA can be derived from theproperties of the series expansions at ERLs for the designatedASV. For the convergent ERLs:

a. f x c x c xlim ( ) :x

MM

MM

0

01

0 10

00

0= + +

++

+

p c p k M, with 0, 0 ; (46)M M k0

00 0= º <⩽

b. f xc

x

c

xlim ( ) :

x

N

N

N

N

1

1= + +

+¥

-¥

- +¥

-¥

¥

¥

¥

p q c N M Nwith . (47)M N N= = +-¥

¥¥

For the divergent ERLs:

a. f xc

x

c

xlim ( ) :

x

N

N

N

N0

01

0

1

0

0

0

0= + +

- - +

-+

p c q q k N, with 1, 0, 0 ;

(48)

N N k00

00 0= = º <- ⩽

b. f x c x c xlim ( ) :x

MM

MM

11= + +

+¥

¥-

¥ -¥

¥¥

¥

p q c M N Mwith . (49)M N M= = +¥¥¥

APPENDIX CEXACT FORMULATION OF ISENTROPIC SOUND SPEED

For the isentropic flow, the differentiation relation ford da( , )z and d de dp( , , )r can be derived subject to theisentropic relation (34) and represented in matrix form as

e A x x A ed d d d a· or · , (50 )1= = -

in which

( )e xdded h

de ddda

b11

, , (50 )rz=

æèççç

öø÷÷÷ = =

æèççç

öø÷÷÷

13


and

Ae e

c. (50 )a

ar r=æ

èççç

ö

ø÷÷÷÷

z

z

Furthermore

B x B A edp d d a· · · , (51 )1= = -

where

B p p b. (51 )a= éë ùûz

Therefore, cs2 can be obtained from Equations (50) and (51) as

cQ Q h

Q, (52)s

2 1 2

3=

+

in which Q1, Q2, and Q3 are defined as

Q p p p X

Q e p p e e Z

Q e e e Y

,

,

,

a a

a a

a a

12

22

32

r r

r r

= - = -

= - = -

= - = -

z z z

z z z

z z z

respectively, where ar r= , e ea = , and p pa = for therelativistic MB flow at equilibrium from Equation (5).Substituting into Equation (52) yields Equation (35). On theother hand, the relation for d de dp( , , )r and ( )X Y Z, ,x x x can bedeveloped by a straightforward calculation subject to theisentropic relation (34), by which one obtains Equation (35)for cs in terms of the ASVs for the CD flow.

REFERENCES

Abramowitz, M. A., & Stegun, I. A. 1972, in Handbook of MathematicalFunctions: with Formulas, Graphs, and Mathematical Tables (New York:Dover)

Adams, B. G. 1988, AcTC, 73, 459Aparicio, J. M. 1998, ApJS, 117, 627Apresyan, L. A. 1979, PADÉ APPROXIMANTS (REVIEW), translated from

IzRad, 22, 653

Baker, G. A., Jr., & Gammel, J. L. 1961, JMAA, 2, 21Banks, T., & Torres, T. J. 2013, arXiv :1307.3689Bender, C. M., & Orszag, S. A. 1978, Advanced Mathematical Methods for

Scientists and Engineers (New York: McGraw-Hill)Bludman, S. A., & van Riper, K. A. 1977, ApJ, 212, 859Cercignani, C., & Kremer, G. M. 2002, The Relativistic Boltzmann Equation:

Theory and Applications, Vol. 22 (Berlin: Birkhäuser)Chandrasekhar, S. 1967, An Introduction to the Study of Stellar Structure (New

York: Dover)Cuyt, A., Tan, J., & Zhou, P. 2006, MaCom, 75, 727deGroot, S. R., van Leeuwen, W. A., & van Weert, Ch. G. 1980, Relativistic

Kinetic Theory, Principles and Applications (Amsterdam, New York:North-Holland)

de Montessus de Ballore, R. 1905, Rend. di Palermo, 19, 1Eggleton, P. P., Faulkner, J., & Flannery, B. P. 1973, A&A, 23, 325Gibson, W. G. 1981, JSP, 26, 333Gil, A., Segura, J., & Temme, N. M. 2007, Numerical Methods for Special

Functions (Philadelphia: SIAM)Gonzhlez-Vera, P., & Paiz, M. J. 1993, ApNM, 11, 385Guillaume, P., & Huard, A. 2000, JCoAm, 121, 197Hakim, R. 2011, Introduction to Relativistic Statistical Mechanics: Classical

and Quantum (Singapore: World Scientific)Hardee, P., & Stone, J. M. 1997, ApJ, 483, 121Huang, K. 1987, Statistical mechanics (New York: Wiley)Landau, L. D., & Lifshitz, E. M. 1980, Statistical physics (Oxford, New York:

Pergamon Press)Mamedov, B. A. 2012, NewA, 17, 353Mathews, W. G. 1971, ApJ, 165, 147McCabe, J. H., & Murphy, J. A. 1976, JIMIA, 17, 233Mignone, A., Plewa, T., & Bodo, G. 2005, ApJS, 160, 199Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. 2010, in NIST

Handbook of Mathematical Functions (Cambridge: Cambridge Univ. Press)Press, W. H., & Vertterling, W. T. 1992, Numerical Recipes in Fortran 77: The

Art of Science Computing (Cambridge: Cambridge Univ. Press)Rezzolla, L., & Zanotti, O. 2013, Relativistic Hydrodynamics (Oxford: Oxford

Univ. Press)Ryu, D., Chattopadhyay, I., & Choi, E. 2006, ApJS, 166, 410Shore, S. N. 2008, Astrophysical Hydrodynamics: An Introduction (2nd ed.;

Weinheim: Wiley-VCH)Stone, J. M., Xu, J., & Hardee, P. 1997, ApJ, 483, 136Taub, A. H. 1948, PhRv, 74, 328Telega, J. J., Tokarzewski, S., & Galka, A. 2000, Acta Applicandae

Mathematica, 61, 295Tooper, R. F. 1969, ApJ, 156, 1075Weinberg, S. 1972, Gravitation and Cosmology: Principles and Applications of

the General Theory of Relativity (New York: Wiley)Xu, J., & Stone, J. M. 1995, ApJ, 454, 172

14


PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY

Documents

Transcript of PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY