1. INTRODUCTION 2. IMPLEMENTATIONS

Transcription

1 THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 125:277È294, 1999 November ( The American Astronomical Society. All rights reserved. Printed in U.S.A. THE ACCURACY, CONSISTENCY, AND SPEED OF FIVE EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS F. X. TIMMES1 AND DAVE ARNETT2 Received 1999 February 5; accepted 1999 May 24 ABSTRACT We compare the thermodynamic properties and execution speed of Ðve independent equations of state. A wide range of temperatures, densities, and compositions are consideredèconditions appropriate for modeling the collapse of a cloud of hydrogen gas (or an exploding supernova) to the outer layers of a neutron star. The pressures and speciðc thermal energies calculated by each equation-of-state routine are reasonably accurate (typically 0.1% error or less) and agree remarkably well with each other, despite the di erent approaches and approximations used in each routine. The derivatives of the pressure and speciðc thermal energies with respect to the temperature and density generally show less accuracy (typically 1% error or less) and more disagreement with one another. Thermodynamic consistency, as measured by deviations from the appropriate Maxwell relations, shows that the Timmes equation of state and the Nadyozhin equation of state achieve thermodynamic consistency to a high degree of precision. The execution speed of the Ðve equation-of-state routinesèevaluated across several di erent machine architectures, compiler options, and modes of operationèdi er dramatically. The Arnett equation of state is the fastest of the Ðve routines, with the Nadyozhin equation of state close behind. Subject headings: equation of state È hydrodynamics È methods: numerical È stars: general È stars: interiors 1. INTRODUCTION Models of stellar events typically require the relationship between various thermodynamic properties over a large span of temperatures, densities, and compositions. Stellar equation-of-stage (EOS) routines are used for conditions ranging from a collapsing hydrogen cloud (or exploding supernova) to a neutron star, so the EOS must be accurate in regions where the electrons have a speed arbitrarily close to the causal limits and an arbitrary degree of degeneracy. With over 109 calls to the EOS being common in two- and three-dimensional hydrodynamic models of stellar phenomena, it is very desirable to have an EOS that is as efficient as possible and yet accurately represents the relevant physics. The purpose of this paper is to compare the accuracy, thermodynamic consistency, and execution speed of Ðve different equation-of-state routines that are used in modeling stellar events. The equation-of-state routines examined in this survey encompass one written to serve as the reference point for the comparisons (hereafter the Timmes EOS), one written by Iben (1963; Iben, Fujimoto, & MacDonald 1992) primarily for evolving models of intermediate- and lowmass stars (hereafter the Iben EOS), one composed by Weaver, Zimmerman, & Woosley (1978) chieñy for evolving models of massive stars (hereafter the Weaver EOS); one summarized by Nadyozhin (1974a, 1974b) and explained in detail by Blinnikov, Dunina-Barkovskaya, & Nadyozhin (1996; hereafter the Nadyozhin EOS), and one developed by Arnett (1969, 1996) for use in multidimensional hydrodynamic simulations (hereafter the Arnett EOS). Our motivation for undertaking this survey is to provide benchmark values for the accuracy and speed of EOS routines that are commonly used for modeling various stellar phenomena and to stimulate further improvement in the algorithms. 1 Center on Astrophysical Thermonuclear Flashes, University of Chicago, Chicago, IL Steward Observatory, University of Arizona, Tucson, AZ These benchmarks allow a full assessment of the Ðve EOS routines and permit practical guidelines to be derived. Stellar EOS routines that were not encompassed by this survey may also beneðt from comparisons with the benchmark values. In 2 we discuss the salient features of the Ðve EOS routines under consideration. The thermodynamic accuracy of the routines is analyzed in 3, how well the routines satisfy thermodynamic consistency is quantiðed in 4, and in 5 we present the results of our timing tests. We summarize our Ðndings in 6 and make some pragmatic suggestions about which EOS routine to use in a given situation. 2. IMPLEMENTATIONS All Ðve EOS routines are based on the physical picture,ïï in which only fundamental species (photons, nuclei, electrons, and positrons) appear. Each routine takes as input the temperature T (in kelvins), density o (in g cm~3), the mean number of nucleons per isotope (A), and the mean charge per isotope (Z). Each EOS routine then returns as output the scalar pressure (in ergs cm~3) and speciðc thermal energy (in ergs g~1) as a sum over the fundamental species: P \ P ] P ] P ] P, tot rad ion ele pos E \ E ] E ] E ] E, (1) tot rad ion ele pos where the subscripts rad,ïï ion,ïï ele,ïï and pos ÏÏ represent the contributions from radiation, nuclei, electrons, and positrons, respectively. Quantities such as the speciðc heats or adiabatic indices can be determined once the partial derivatives of the pressure and speciðc thermal energy with respect to the density and temperature are known. The adiabatic indices, for example, are often vital input for multidimensional hydrodynamic algorithms. In cases where an energy equation instead of a temperature equation is solved (as in most hydrodynamic algorithms), the temperature is usually

2 278 TIMMES & ARNETT Vol. 125 obtained by iteration, and the thermodynamic derivatives must be available in order to implement efficient iteration schemes. Similarly, stellar evolutionary algorithms require the thermodynamic derivatives for convergence. Checking that an EOS routine numerically satisðes various thermodynamic identities also demands that the derivatives be available. For these reasons, each of the Ðve EOS routines returns the partial derivatives of the pressure and speciðc thermal energy with respect to the density and temperature: LP tot LT LP tot Lo LE tot LT LE tot Lo,,,, (2) o T o T The radiation portion of each routine is simple, always a blackbody in local thermodynamic equilibrium: P \ at 4 rad 3, E rad \ 3P rad, (3) o where a is related to the Stefan-Boltzmann constant p \ ac/4, and c is the speed of light. The ion section of each routine is also simple, always an ideal gas: N \ N Ao ion A, P ion \ N ion kt, E ion \ 3 P ion 2 o, (4) where N is the ion number density, N is AvogadroÏs ion A number, and k is BoltzmannÏs constant. Each routine follows the formalism of a noniteracting Fermi gas for the electrons and positrons. The number density of free electrons N and positrons N is ele pos N \ 8nJ2 m3 c3b3@2[f (g, b) ] F (g, b)], ele h3 e 1@2 3@2 N \ 8nJ2 m3 c3b3@2 pos h3 e ] [F ([g [ 2/b, b) ] bf ([g [ 2/b, b)], (5) 1@2 3@2 where h is PlanckÏs constant, m is the electron rest mass, e b \ kt /(m c2) is the relativity parameter, g \ k/kt is the e normalized chemical potential energy k for electrons, and F (g, b) is the Fermi-Dirac integral k P = xk(1 ] 0.5bx)1@2 dx F (g, b) \ k exp (x [ g) ] 1. (6) 0 In this formulation the normalized chemical potential g has the rest mass energy of the electrons subtracted out, so g is the kinetic chemical potential (the u in Appendix eq. [B54] of Arnett 1996). This means that the positron chemical potential must have the rest-mass terms appear explicitly, g \[g[2/b, as it does in equation (5). Note that b as deðned pos here is the multiplicative inverse of the b used in Chandrasekhar (1939) and Arnett (1996), and of the z in Fowler & Hoyle (1964); this allows us to avoid the clutter of many negative exponents. The terms Fermi-Dirac,ÏÏ generalized Fermi-Dirac,ÏÏ and Fermi ÏÏ integral have not received uniform usage in the literature. This paper will use the term Fermi-Dirac ÏÏ integral for the three-parameter function in equation (6), and reserve the term Fermi ÏÏ integral as a special case (b \ 0) of the Fermi-Dirac integral. Note that equation (2) implies that the derivatives of the various Fermi-Dirac integrals with respect to g and b must be available. For complete ionization, the number density of free electrons in the matter is and charge neutrality requires N ele,matter \ Z A N a o \ ZN ion, (7) N ele,matter \ N ele [ N pos. (8) Solving equation (8) determines the normalized chemical potential g, which was the only unknown. Such a solution fulðlls the chemical potentialïs role as the Lagrange multiplier that was originally introduced to constrain the distribution function to have the correct number of particles. Solving equation (8) in practice means Ðnding its root g. Newton-Raphson or Brent iteration are common methods for performing this one-dimensional root Ðnd. Once g is known from the solution of equation (8), the pressure and speciðc thermal energy due to free electrons and positrons is P ele \ 16nJ2 3h3 P pos \ 16nJ2 3h3 E ele \ 8nJ2 oh3 C m4 c5b5@2 F (g, b) ] 1 e 3@2 2 bf (g, b)d 5@2 m e 4 c5b5@2 C ] F ([g [ 2/b, b) ] 1 3@2 2 bf ([g [ 2/b, b)d 5@2 m e 4 c5b5@2[f 3@2 (g, b) ] bf (g, b)] 5@2 E \ 8nJ2 pos oh3 m e 4 c5b5@2[f ([g [ 2/b, b) 3@2 ] bf 5@2 ([g [ 2/b, b)] ] 2m e c2n pos o. (9) Many authors have examined the thermodynamics of an ideal fermion-antifermion gas and derived useful approximations to its equation of state (see Cox & Giuli 1968; Chiu 1968; Eggleton, Faulkner, & Flannery 1973; Bludman & Van Riper 1977; Blinnikov et al. 1996). It is not our intention to review the subject or to provide new approximations, only to elucidate the key di erences in how each of the Ðve EOS routines under consideration implements the root Ðnd of equation (8) and the integrals of equation (5) and (9). The Timmes EOS routine was designed for maximum accuracy and will serve as the reference point for comparisons with the other EOS routines. Evaluations of the Fermi-Dirac integrals (eq. 6), along with their derivatives with respect to g and b, are calculated to at least 18 signiðcant Ðgures with the quadrature schemes of Aparicio (1998). That is, the Fermi-Dirac integrals and their derivatives are exact in IEEE 64 bit arithmetic (16 signiðcant Ðgures). Iteration with a Newton-Raphson scheme is used to solve equation (8) for the normalized chemical potential g to at least 15 signiðcant Ðgures. The partial derivatives of equation (2) are formed analytically. As a check, these partial derivatives were calculated numerically by second-orderè accurate di erence equations, and the values agreed with the analytical results to the precision expected of the di er-

3 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 279 ence equations. Finally, the 1986 recommended values of the fundamental physical constants (Cohen & Taylor 1987) are entered to their published precision. The Iben EOS routine uses an approximation due to Eggleton et al. (1973) that implements global Ðtting formulae for the Fermi-Dirac integrals. This approximation is based on a polynomial of two variables which has a multiplier function that gives the correct asymptotic behavior for very small and very large values of g and b. Contributions from positrons in equations (5) and (9) are neglected by the Iben EOS. Thus, one expects some inaccuracies when pair production makes a signiðcant contribution to the pressure and speciðc thermal energy. The normalized chemical potential g is calculated to at least six signiðcant Ðgures by Newton-Raphson iteration of equation (8), with N \ 0. The derivatives of the thermodynamic quantities pos with respect to g and b are calculated by Ðrst-orderÈaccurate Ðnite di erence formula, so the partial derivatives of equation (2) are formed from combinations of Ðnite di erence and analytical terms. The Weaver EOS routine uses an approximation by Divine (1965) that modiðes Equations (5) and (9) in such a way that only the Fermi integrals, instead of the Fermi- Dirac integrals, are needed. This approximation is based on a third-order rational function which becomes exact in the four relativistic and degenerate limits. The maximum error of the Divine approximation, which occurs in the intermediate regime (E D kt D m c2), is less than 0.3% (Divine Fermi e 1965; Chiu 1968; Figs. 1 and 3 of this paper). Evaluation of the Fermi integrals is accomplished using a cubic spline interpolant on stored tablesï values. The tables are stored to seven signiðcant Ðgures and go from g \[4.0 to ]20.0 in 67 unequally spaced increments. Second- or third-orderè accurate expansions for the thermodynamic quantities are used when g falls outside these table limits. The normalized chemical potential g is calculated to at least six signiðcant Ðgures by Newton-Raphson iteration of equation (8). The partial derivatives of equation (2) are formed analytically. The Nadyozhin EOS routine uses polynomial or rational functions to evaluate the thermodynamic quantities. The temperature-density plane is decomposed into Ðve regions (see Fig. 12 of Blinnikov et al. 1996), with di erent expansions or Ðtting functions applied to each region. The methods used in the Ðve regions are (1) perfect gas approximation with the Ðrst-order corrections for degeneracy, (2) expansions of the half-integer Fermi-Dirac functions, (3) ChandrasekharÏs (1939) expansion for a degenerate gas, (4) relativistic asymptotics, and (5) Gaussian quadrature for the thermodynamic quantities. The perturbation expansions, asymptotic relations, and Ðtting functions for the Ðve regions are combined, with extraordinary care given to making sure that transitions between the regions are continuous, smooth, and thermodynamically consistent. All of the partial derivatives (eq. 2) are obtained analytically. The Nadyozhin EOS gains elegance and execution efficiency by using new, analytical expressions for a partially relativistic electron-positron gas (Blinnikov et al. 1996). The Arnett EOS routine uses table lookup to evaluate the electron-positron thermodynamic quantities. The Fermi- Dirac integrals (eq. 6) and the root Ðnd for the normalized chemical potential g (eq. 8) are computed o -line. The target accuracy of the integrations, and of the iteration for the chemical potential, is one part in The results of the o -line calculation are a table of the electron-positron pressure and speciðc thermal energy as a function of the density and temperature. The table stores the results to nine signið- FIG. 1.ÈPressure vs. density for various temperatures and a pure 12C (Z \ 6,A \ 12) composition

4 280 TIMMES & ARNETT Vol. 125 FIG. 2.ÈAbsolute value of the di erence from the exact Timmes EOS for the scalar pressure cant Ðgures, and operates in the range 10~4 \ (Z/A)o\3 ]1010 gcm~3 in 117 equally spaced (in the logarithm) increments and 105 \ T \ 1010 in 101 equally spaced (in the logarithm) steps. The limits and increments of the table were chosen to provide sufficient accuracy in the thermodynamic variables. Searches through the table are avoided by computing the table indices from the logarithmic values of any given (T, oz/a) pair (i.e., the table is hashed). Notice that this two-dimensional scheme for the electron-positron plasma is valid for any composition characterized by Z/A; separate planes for each Z/A are not necessary. Of course, like the other EOS routines, all four variables (T, o, Z,A) are needed for a complete evaluation. Interpolation in the table is done with bicubic polynomials, which returns the

5 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 281 electron-positron thermodynamic quantities. The electronpositron thermodynamic derivatives are constructed from analytic di erentiation of the interpolating polynomials. Second- or third-orderèaccurate expansions are used when the temperature and density fall outside the table limits, and in these regions the electron-positron thermodynamic quantities and their derivatives are analytic. Most of the Ðve EOS routines provide options for including corrections due to partial ionization, Coulomb interactions, and the onset of ion degeneracy. These corrections terms are turned o in our comparisons, chieñy because some of the EOS routines do not provide options for including them. An additional reason for turning these corrections o is that the physics used to describe these e ects in the EOS routines that include them is slightly di erent in each routine. This paper seeks to compare di erent implementations of the same basic physics, not to compare di erent physics. 3. THERMODYNAMIC ACCURACY The pressure and speciðc thermal energy, along with their derivatives with respect to density and temperature, were computed by each of the Ðve EOS routines for temperatures between 105 and 1010 and densities from 10~4 to 1011 g cm~3. Temperatures smaller than this lower bound may require the addition of partial ionization models, while densities larger than this upper bound may require the addition of ion degeneracy models. Compositions of pure hydrogen, pure 4He, solar (1.0Z ), 0.1Z, 2.0Z, pure 12C, and pure _ 30Si were used in comparing the output of the Ðve EOS routines. It is important to note that the accuracy, thermodynamic consistency, and speed of each EOS routine is independent of the composition. Accordingly, only the results for a pure 12C composition are reported in this paper. The conclusions drawn from this particular composition, however, are valid for any composition. Each routine was executed with IEEE 64 bit arithmetic, which has 16 signiðcant Ðgures. Other 64 bit schemes (like the Cray Ñoating point) may have fewer (or more) signiðcant Ðgures. Figure 1 shows the scalar pressure as a function of the density. Each group of points corresponds to the labeled temperature. The circles, crosses, squares, triangles, and pentagons represent the pressure as given by the Timmes, Iben, Weaver, Nadyozhin, and Arnett EOS routines, respectively. Almost all the symbols overlap in Figure 1, indicating that on logarithmic scales where the density changes by 15 orders of magnitude, each EOS routine calculates nearly the same pressure. At temperatures greater than D108 and densities less than 107 gcm~3, the Iben EOS does not agree with the other EOS routines because it does not include contributions from positrons. The pressure relative to the (exact) Timmes EOS is shown in Figure 2. Symbols in Figure 2 and in all the remaining Ðgures have the same meaning as the symbols in Figure 1. The error made by the Iben EOS in regions where positrons are not important is about 0.1% or less, consistent with values found by Eggleton et al. (1973). At D109 and densities less then 107 g cm~3 the typical error increases to about 10%, while for D1010 the typical error is about a factor of 3. Again, the chief reason for this behavior is that the Iben EOS neglects positrons. Note that the errors for the Iben EOS exceed the size of the y-axis at the highest temperatures, causing the curves to suddenly ÏÏ appear as the density increases. The typical error made by the Weaver FIG. 3.ÈSpeciÐc thermal energy vs. density for various temperatures and a pure 12C composition

6 282 TIMMES & ARNETT Vol. 125 FIG. 4.ÈAbsolute value of the di erence from the Timmes EOS for the speciðc thermal energy EOS is about 0.01%, with a maximum error of about 0.3% occurring in regions where the electron-positron gas is partially relativistic and partially degenerate. This is consistent with what other authors have found when the Divine approximation is used (Divine 1965; Chiu 1968). Discontinuities in the error made by the Weaver EOS are caused by transitions to regions where di erent Fermi integral tables or di erent expansions are used. Typical errors made by the Nadyozhin EOS are about 0.01% or less, with the maximum error of about 0.1% located in regions where

7 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 283 FIG. 5.ÈPartial derivative of the pressure with respect to the density at constant temperature transitions between di erent expansions or Ðtting formulae take place. Of the four EOS routines shown in Figure 2, the Nadyozhin EOS has the smallest errors and the most uniform error distribution. The error made by the Arnett EOS is about 0.03% or less, with the maximum error of 0.1% located in regions that are near the boundaries of the table. Periodic structures in the Arnett EOS error distribution at high densities are due not to inaccuracies in the table values but rather to the structure of the interpolating bicubic polynomial. Figure 3 shows the speciðc thermal energy as a function of the density. Most of the symbols in Figure 3 overlap, indicating that each EOS routine calculates roughly the same speciðc thermal energy (at least on wide-ranging logarithmic scales). The speciðc thermal energy relative to the (machine precision) Timmes EOS is shown in Figure 4. The typical errors and the locations of maximum error are quite similar to those for the errors in the pressure, hence the analysis of Figures 1 and 2 applies in Figures 3 and 4. Figures 1È4 demonstrate that the pressures and speciðc thermal energies calculated by each EOS routine are reasonably precise and agree with one another remarkably well, despite the di erent implementations of the same physics. A factor that limits the level of agreement is the precision with which the fundamental physical constants are entered and manipulated. In regions where radiation terms dominate the pressure, for example, values of the constant a (eq. 3) that di er in the fourth signiðcant Ðgure can only result in a relative agreement of 0.01%. As noted in 3, the Timmes EOS uses the 1986 recommended values of the fundamental physical constants to their published precision. All the other EOS routines often enter the fundamental physical constants, or various combinations of them, to 3È5 signiðcant Ðgures. The relative di erence in the fundamental constants accounts for some, but not all, of the errors shown in Figures 2 and 4. Substantial e ort would be required to make all the EOS routines use the same values of the physical constants everywhere, and this e ort was not attempted. The derivatives of equation (2) and their associated relative errors are shown in Figures 5È12. All of the EOS routines calculate roughly the same derivative values, at least on the global logarithmic axes of Figures 5, 7, 9, and 11. More interesting behavior is evident in Figures 6, 8, 10, and 12 for the errors each routine makes relative to the Timmes EOS. The typical error made by the Iben EOS in regions where positrons are not important is about 1%, sometimes less. When contributions from positrons are signiðcant, the error is much larger. The overall distribution of the relative error made by the Iben EOS is relatively Ñat. Again, the relative errors of the Iben EOS exceed the y-axis scale at the highest temperatures, which causes the data points to suddenly appear as the density increases. The typical error made by the Weaver EOS is about 0.01%, with a maximum error of about 0.1% occurring mainly in regions where the electron-positron gas is partially relativistic and partially degenerate. Apparent discontinuities in the error made by the Weaver EOS are caused by the error changing signs and the plots showing the absolute value of the error. Errors made by the Nadyozhin EOS in the four derivatives are about 0.01% or less, with the maximum error of about 0.4% located in regions where transitions between di erent expansions or Ðtting formulae take place. Exclusive of these spikes in the error at transition regions, the Nadyozhin

8 284 TIMMES & ARNETT Vol. 125 FIG. 6.ÈAbsolute value of the di erence from the Timmes EOS for the derivative of the pressure with respect to the density EOS has the smallest errors in the derivatives. The error made by the Arnett EOS in the derivatives is about 0.1%, with the maximum error of 3% located in regions that are near the boundaries of the table. The chief reason that errors in the derivatives are larger than in the thermodynamic quantities is that the thermodynamic derivatives are determined from the derivative of the interpolating bicubic polynomial. A periodic structure in the distribution of errors, especially at high densities where the electrons are degenerate, is due to the interpolant showing its structure.

9 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 285 FIG. 7.ÈPartial derivative of the pressure with respect to the temperature at constant density In general, the derivative quantities returned by each of these four EOS routines show less precision than the integrated quantities. This is due mainly to the nature of the approximations or interpolating procedures used by each routine. 4. THERMODYNAMIC CONSISTENCY The Ðrst law of thermodynamics is an exact di erential, which implies P \ o2 LE Lo LE LT o T ] T LP LT \ T LS LT, (10) o, (11) o [ LS \ 1 LP, (12) Lo o2 LT T o where S is the entropy (in ergs g~1 ~1). An equation of state is thermodynamically consistent if all three of these relations are true. Thermodynamic inconsistency may manifest itself in the unphysical buildup (or decay) of the entropy (or temperature) during numerical simulations of what should be an adiabatic Ñow. Models of events that are sensitive to the entropy (e.g., core-collapse supernovae) may su er inaccuracies if thermodynamic consistency is signiðcantly violated over sufficiently long timescales. Since all Ðve EOS routines return the partial derivatives needed to evaluate equation (10), how well each routine satisðes this thermodynamic consistency relation can be quantiðed. Figure 13 shows the deviation that each EOS routine makes in satisfying equation (10). The smaller the deviation, with zero deviation being the perfect case, the closer the equation of state comes to satisfying this thermodynamic consistency relation. Figure 13 demonstrates that the Timmes EOS satisðes equation (10) to the limiting precision of IEEE 64 bit arithmetic over the entire temperature-density plane under consideration. The Nadyozhin EOS generally satisðes the consistency constraint to a high degree of precision. Abrupt changes in the deviation are due to traversing the various regions into which the Nadyozhin EOS decomposes the temperature-density plane. The Iben EOS tends to have a larger deviation than the Nadyozhin EOS, but avoids any abrupt changes in the deviation. Deviations of the Iben EOS are not plotted for temperatures of 109 and 1010 in Figure 13, since positrons are neglected and the accuracy of the returned thermodynamic quantities becomes degraded (an EOS can be consistent yet inaccurate in some regions). The Weaver EOS tends to have even larger deviations from satisfying equation (10), with the maximum deviations occurring in regions of hard degeneracy where subtraction of two large and nearly equal terms determines some of the partial derivatives. The Arnett EOS generally has the largest deviations from satisfying equation (10). A denser table would probably decrease the deviations. Table lookup schemes tend to be more susceptible to thermodynamic inconsistencies than other schemes, whether the thermodynamic partial derivatives are formed as the partial derivatives of the interpolating polynomials (as is the case here) or an interpolation is performed on a table of the partial derivatives (Swesty 1996). Only the Nadyozhin EOS and the Timmes EOS return the entropy derivatives necessary to evaluate the thermodynamic consistency relations in equations (11) and (12). Figure 14 shows the deviation that these two EOS routines make in satisfying equation (11), while the deviation from achiev-

10 286 TIMMES & ARNETT Vol. 125 FIG. 8.ÈAbsolute value of the di erence from the Timmes EOS for the derivative of the pressure with respect to the temperature ing equation (12) is shown in Figure 15. These Ðgures indicate that the Timmes EOS satisðes the corresponding consistency relations to the limiting precision of IEEE 64 bit arithmetic. In some regions of the temperature-density plane, the deviations made by the Nadyozhin EOS are as small as the deviations made by the Timmes EOS. Again, discontinuities in the deviations made by the Nadyozhin EOS are due to crossing into regions where di erent expansions or asymptotic relations are used. Overall, Figures 13È15 suggest that the Timmes EOS and Nadyozhin EOS achieve thermodynamic consistency by meeting equations (10)È(12) to a high degree of precision. It is difficult to give a complete assessment of the thermodynamic consistency of the Iben EOS, the Weaver EOS, and the

11 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 287 FIG. 9.ÈPartial derivative of the speciðc thermal energy with respect to the density at constant temperature Arnett EOS, since these routines do not return the requisite entropy derivatives. However, if the behavior shown in Figure 13 is indicative, then these equations of state are not as consistent as either the Timmes EOS or the Nadyozhin EOS. 5. EXECUTION EFFICIENCY The speed of each EOS was evaluated by calling it 108 times in ordered, random, and constant entropy sweeps. An ordered sweep consists of setting the temperature to the smallest value in the range considered (105 ) and then looping through 1000 density points, starting from the smallest value (10~4 g cm~3) and Ðnishing on the largest value (1011 gcm~3), with data points evenly spaced in the logarithm. This is repeated for 104 temperature points, Ðnishing on the largest value (1010 ) in evenly spaced logarithmic intervals. In this manner, the entire temperature-density plane under consideration was uniformly sampled. A random sweep consisted of choosing an arbitrary temperature and density which was uniformly distributed in the range considered. This type of sweep was chosen to minimize any speed advantage an EOS routine might gain by using information from a previously computed or neighboring point. An entropy sweep consisted of setting the temperature to the smallest value and then looping through 104 density points chosen in such a way that the entropy (expression) remained constant. This is repeated for 104 temperature points, Ðnishing on the largest value in the range considered. This type of sweep was chosen as part of the timing tests because most of a starïs life (particularly a massive starïs) is spent evolving at roughly constant entropy. The total CPU time spent executing each type of sweep was divided by the 108 calls to obtain the number of CPU seconds per call. The timing tests were run on seven di erent serial computers, six UNIX workstations and one LINUX PC. Each of the computers had a di erent CPU clock speed (180È 450 MHz), bus clock speed (30È400 Mbyte s~1), and cache memory size (0.512È4 Mbytes). Each EOS routine was compiled under Fortran 77 and Fortran 90. When possible, the compilation was performed with one of eight di erent compiler option sets, from a set that requested no code optimization to a set that requested routines to be in-lined, do-loops to be unrolled, and aggressive code optimization. The absolute speed of each EOS routine depended, obviously, on the machine architecture and compiler options employed. These dependences can be minimized, and meaningful comparisons made, by comparing the relative speed of each EOS routine. For this reason, the results of the timing tests shown in Tables 1 and 2 were normalized to the Arnett EOS. It bears repeating that the relative speed rankings shown in Tables 1 and 2 are nearly the same for any given serial machine; the numbers in the tables are not some sort of average across the various machines. Table 1 shows the relative timing results when each EOS routine operates in serial mode. In serial mode, each routine operates on a single temperature, density, and composition point. A separate call is required for each input. Table 2 shows the relative timing results when each EOS routine operates in pipeline mode. In pipeline mode, each routine operates on temperature, density, and composition arrays. A single call is required for an array of input values, reward-

12 288 TIMMES & ARNETT Vol. 125 FIG. 10.ÈAbsolute value of the di erence from the Timmes EOS for the derivative of the speciðc thermal energy with respect to the density ing routines that make efficient use of the memory cache. All the values in Tables 1 and 2 have been normalized to the Arnett EOS for ordered sweeps in serial mode. Tables 1 and 2 indicate that the speed of the Nadyozhin EOS and the Arnett EOS routines increases by a factor of 2È4 when operated in pipeline mode rather than serial mode, because use of the computerïs cache memory is more efficient. These two EOS routines executed ordered sweeps about 40% faster than random sweeps, and about 20% faster than entropy sweeps. The chief reason for this behavior is that information for neighboring points is located in close proximity in physical memory. Since ordered sweeps calculate the EOS for neighboring points, and random sweeps calculate the EOS for widely scattered points, the ordered sweep is more likely than the random sweep to access data already loaded into the processor cache rather

13 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 289 FIG. 11.ÈPartial derivative of the speciðc thermal energy with respect to the temperature at constant density TABLE 1 RELATIVE TIMINGS IN SERIAL MODEa TYPE OF TEMPERATURE AND DENSITY SWEEP EOS Ordered Random Entropy Timmes Iben Weaver Nadyozhin Arnett a CPU time per equation of state calls for each EOS operating in serial mode. All values have been normalized to the Arnett EOS for ordered sweeps in serial mode. All values are generally independent of the machine architecture and compiler options used. TABLE 2 RELATIVE TIMINGS IN PIPELINE MODEa TYPE OF TEMPERATURE AND DENSITY SWEEP EOS Ordered Random Entropy Timmes Iben Weaver Nadyozhin Arnett a CPU time per equation of state calls for each EOS operating in pipeline mode. All values have been normalized to the Arnett EOS for ordered sweeps in serial mode. All values are generally independent of the machine architecture and compiler options used. than having to access these data from the slower main memory. This reduction in the time required to access information from memory translates into a faster overall execution speed. Tables 1 and 2 show that the speed of the Timmes EOS, Iben EOS, and Weaver EOS is about the same in serial and pipeline modes for all sweep types. The reason for this behaviour is that these EOS routines perform the root Ðnd of equation (8) in-line, which consumes the majority of the CPU time for any given temperature and density input point. Tables 1 and 2 suggest that the Arnett EOS is the fastest of the Ðve routines, with the Nadyozhin EOS a close second. The Iben EOS and the Weaver EOS are slower, chieñy because they perform the root Ðnd of equation (8) in-line. The Iben EOS is slower than the Weaver EOS because some of the derivatives are computed by di erence formula, which requires additional calls to the slowest part of the routine. The Timmes EOS, not surprisingly, is the slowest, since it was designed to forsake speed in favor of accuracy. These relative speed rankings are nearly the same for any given serial machine. 6. SUMMARY AND SUGGESTIONS The accuracy, thermodynamic consistency, and execution speed of Ðve EOS routines that are suitable for models of a wide variety of stellar phenomena have been assessed. Despite the di erent implementations of the same basic physics by each of the Ðve EOS routines, their calculated pressures and speciðc thermal energies were within 0.3% of the exact results, while the derivatives were within 0.6% of the exact results. Two of the EOS routines achieved ther-

14 290 TIMMES & ARNETT Vol. 125 FIG. 12.ÈAbsolute value of the di erence from the Timmes EOS for the derivative of the speciðc thermal energy with respect to the temperature modynamic consistency to a high degree of precision, while a complete assessment of the thermodynamic consistency of the other three routines was not possible since the entropy derivatives were missing. The results of this survey (the Ðgures and tables) permit a few pragmatic suggestions to be made. If absolute accuracy and thermodynamic consistency are the primary concern, then there is no substitute for obtaining the Fermi-Dirac functions to high precision, reducing the uncertainty in the value of g for the root Ðnd to the limits of the arithmetic, having analytic partial derivatives, and entering values of the fundamental physical constants to their recommended

15 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 291 FIG. 13.ÈDeviation of the thermodynamic relation P \ o2(le/lo) o ] T (LP/LT ) o. The smaller the deviation, with zero deviation being the perfect case, T o the closer the equation of state comes to satisfying thermodynamic consistency. The Iben EOS is not plotted for the largest temperatures since it is inaccurate when positrons make signiðcant contributions. precision. This type of approach was implemented in the Timmes EOS. The downside of this approach is a signiðcant increase in the CPU resources needed to model a stellar event. Given the many and perhaps large uncertainties that a ect the results of stellar models (the treatment of convection, residual disagreement on key nuclear reaction rates, precision of initial conditions), any of the other four EOS routines provides sufficient accuracy at less expense. If execution speed is the primary concern, then a table lookup scheme operating in pipeline mode is the best

16 292 TIMMES & ARNETT Vol. 125 FIG. 14.ÈDeviation of the thermodynamic relation (LE/LT ) o \ T (LS/LT ) o. The smaller the deviation, with zero deviation being the perfect case, the o o closer the equation of state comes to satisfying thermodynamic consistency. Only the Nadyozhin EOS and the Timmes EOS return the entropy derivatives needed to evaluate this expression. choice. The Arnett EOS is a good example of this type of approach, and was easily the fastest of the Ðve EOS routines. A downside of this approach is that the tables and interpolants require management to ensure that adequate levels of accuracy and thermodynamic consistency are maintained. In the course of this survey, changes were made to the Arnett EOS that signiðcantly improved its accuracy and thermodynamic consistency. A possible advantage of the table lookup approach is the ability to embed additional physics in the table without a ecting the execution speed.

17 No. 1, 1999 EQUATIONS OF STATE FOR STELLAR HYDRODYNAMICS 293 FIG. 15.ÈDeviation of the thermodynamic relation [(LS/Lo) o \ (1/o2)(LP/LT ) o. The smaller the deviation, with zero deviation being the perfect case, T o the closer the equation of state comes to satisfying thermodynamic consistency. Only the Nadyozhin EOS and the Timmes EOS return the entropy derivatives needed to evaluate this expression. When an optimal balance between accuracy, thermodynamic consistency, and speed is desirable, then the Nadyozhin EOS is a very good choice. This work has been supported by the Department of Energy under grant B to the Center on Astrophysical Thermonuclear Flashes at the University of Chicago, and by grant FG03-98DP00214/A001 at the University of Arizona. The authors thank the referee Raphael Hix for his review; several of his suggestions improved the quality and accuracy of this paper. Most of the equation-ofstate routines may be obtained by contacting the authors.

18 294 TIMMES & ARNETT Aparicio, J. M. 1998, ApJS, 117, 627 Arnett, D. 1969, Ap&SS, 5, 180 ÈÈÈ Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present (Princeton: Princeton Univ. Press) Blinnikov, S. I., Dunina-Barkovskaya, N. V., & Nadyozhin, D , ApJS, 106, 171 Bludman, S. A., & Van Riper,. A. 1977, ApJ, 212, 859 Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure (Chicago: Univ. Chicago Press) Chiu, H.-Y. 1968, Stellar Physics, Vol. 1 (Waltham: Blaisdell) Cohen, E. R., & Taylor, B. N. 1987, J. Res. NBS, 92, 2 REFERENCES Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure (New York: Gordon & Breach) Divine, N. 1965, ApJ, 142, 1652 Eggleton, P. P., Faulkner, J., & Flannery, B. P. 1973, A&A, 23, 325 Fowler, W. A., & Hoyle, F. 1964, ApJS, 91, 1 Iben, I., Jr. 1963, ApJ, 138, 452 Iben, I., Jr., Fujimoto, M. Y., & MacDonald, J. 1992, ApJ, 388, 521 Nadyozhin, D a, Nauchnye informatsii Astron., Soviet USSR, 32, 3 ÈÈÈ. 1974b, Nauchnye informatsii Astron., Soviet USSR. 33, 117 Swesty, D. 1996, J. Comput. Phys., 127, 118 Weaver, T. A., Zimmerman, G. B., & Woosley, S. E. 1978, ApJ, 225, 1021 Note added in proof.èf. X. Timmes & D. Swesty (ApJ, in press [1999]) develop and analyze an electron-positron EOS based on a biquintic interpolation of the Helmholtz free energy. Compared to the Ðve EOS routines surveyed above, the Helmholtz EOS is more accurate than all but the Timmes EOS, is more thermodynamically consistent than any of them (in some cases an average of 8 orders of magnitude more consistent), and is faster than any of the Ðve EOS routines tested.