# Equation of State Unit

## Introduction

The Eos unit implements the equation of state needed by the hydrodynamics and nuclear burning solvers. The function physics/Eos/Eos provides the interface for operating on a one-dimensional vector. The same interface can be used for a single cell by reducing the vector size to 1. Additionally, this function can be used to find the thermodynamic quantities either from the density, temperature, and composition or from the density, internal energy, and composition. For user’s convenience, a wrapper function (physics/Eos/Eos_wrapped) is provided, which takes a section of a block and translates it into the data format required by the physics/Eos/Eos function, then calls the function. Upon return from the physics/Eos/Eos function, the wrapper translates the returned data back to the same section of the block.

Four implementations of the (Eos) unit are available in the current release of Flash-X: Gamma which implements a perfect-gas equation of state; Gamma/RHD which implements a perfect-gas equation taking relativistic effects into account; Multigamma which implements a perfect-gas equation of state with multiple fluids, each of which can have its own adiabatic index ($$\gamma$$); and Helmholtz which uses a fast Helmholtz free-energy table interpolation to handle degenerate/relativistic electrons/positrons and includes radiation pressure and ions (via the perfect gas approximation).

As described in previous sections, Flash-X evolves the Euler equations for compressible, inviscid flow. This system of equations must be closed by an additional equation that provides a relation between the thermodynamic quantities of the gas. This relationship is known as the equation of state for the material, and its structure and properties depend on the composition of the gas.

It is common to call an equation of state (henceforth EOS) routine more than $$10^9$$ times during a two-dimensional simulation and more than $$10^{11}$$ times during the course of a three-dimensional simulation of stellar phenomena. Thus, it is very desirable to have an EOS that is as efficient as possible, yet accurately represents the relevant physics. While Flash-X is capable of using any general equation of state, we discuss here the three equation of state routines that are supplied: an ideal-gas or gamma-law EOS, an EOS for a fluid composed of multiple gamma-law gases, and a tabular Helmholtz free energy EOS appropriate for stellar interiors. The two gamma-law EOSs consist of simple analytic expressions that make for a very fast EOS routine both in the case of a single gas or for a mixture of gases. The Helmholtz EOS includes much more physics and relies on a table look-up scheme for performance.

## Gamma Law and Multigamma

Flash-X uses the method of Colella & Glaz (1985) to handle general equations of state. General equations of state contain 4 adiabatic indices (Chandrasekhar 1939), but the method of Colella & Glaz parameterizes the EOS and requires only two of the adiabatic indices The first is necessary to calculate the adiabatic sound speed and is given by

$\gamma_1 = \frac{\rho}{P}\frac{\partial P}{\partial \rho} \; .$

The second relates the pressure to the energy and is given by

$\label{Eqn:game}\gamma_4 = 1 + \frac{P}{\rho\epsilon} \; .$

These two adiabatic indices are stored as the mesh-based variables GAMC_VAR and GAME_VAR. All EOS routines must return $$\gamma_1$$, and $$\gamma_4$$ is calculated from [Eqn:game].

The gamma-law EOS models a simple ideal gas with a constant adiabatic index $$\gamma$$. Here we have dropped the subscript on $$\gamma$$, because for an ideal gas, all adiabatic indices are equal. The relationship between pressure $$P$$, density $$\rho$$, and specific internal energy $$\epsilon$$ is

$\label {Eqn:eos2b} %{P = {N_a \ k \over \bar{A}} \rho T} P = \left(\gamma - 1\right)\rho\epsilon~. %\epsilon = {1 \over \gamma - 1} \ {P \over \rho}$

We also have an expression relating pressure to the temperature $$T$$

$\label {Eqn:eos2a} P = \frac{N_a k}{\bar{A}} \rho T ~,$

where $$N_a$$ is the Avogadro number, $$k$$ is the Boltzmann constant, and $$\bar{A}$$ is the average atomic mass, defined as

$\frac{1}{\bar{A}} = \sum_{i}\frac{X_{i}}{A_{i}}~,$

where $$X_i$$ is the mass fraction of the $$i$$th element. Equating these expressions for pressure yields an expression for the specific internal energy as a function of temperature

$\epsilon = \frac{1}{\gamma - 1} \frac{N_a k} {\bar{A}} T~.$

The relativistic variant of the ideal gas equation is explained in more detail in .

Simulations are not restricted to a single ideal gas; the multigamma EOS simulations with several species of ideal gases each with its own value of $$\gamma$$. In this case the above expressions hold, but $$\gamma$$ represents the weighted average adiabatic index calculated from

$\frac{1}{\left(\gamma - 1\right)} = \bar{A}\sum_{i}\frac{1}{\left(\gamma_{i} - 1\right)}\frac{X_{i}}{A_{i}}~.$

We note that the analytic expressions apply to both the forward (internal energy as a function of density, temperature, and composition) and backward (temperature as a function of density, internal energy and composition) relations. Because the backward relation requires no iteration in order to obtain the temperature, this EOS is quite inexpensive to evaluate. Despite its fast performance, use of the gamma-law EOS is limited, due to its restricted range of applicability for astrophysical problems.

## Helmholtz

The Helmholtz EOS provided with the Flash-X distribution contains more physics and is appropriate for addressing astrophysical phenomena in which electrons and positrons may be relativistic and/or degenerate and in which radiation may significantly contribute to the thermodynamic state. Full details of the Helmholtz equation of state are provided in Timmes & Swesty (1999). This EOS includes contributions from radiation, completely ionized nuclei, and degenerate/relativistic electrons and positrons. The pressure and internal energy are calculated as the sum over the components

$\label {Eqn:eos3a} P_{\rm tot} = P_{\rm rad} + P_{\rm ion} + P_{\rm ele} + P_{\rm pos} + P_{\rm coul}$
$\label {Eqn:eos3b} \epsilon_{\rm tot} = \epsilon_{\rm rad} + \epsilon_{\rm ion} + \epsilon_{\rm ele} + \epsilon_{\rm pos} + \epsilon_{\rm coul} \; .$

Here the subscripts “rad,” “ion,” “ele,” “pos,” and “coul” represent the contributions from radiation, nuclei, electrons, positrons, and corrections for Coulomb effects, respectively. The radiation portion assumes a blackbody in local thermodynamic equilibrium, the ion portion (nuclei) is treated as an ideal gas with $$\gamma \, = \, 5/3$$, and the electrons and positrons are treated as a non-interacting Fermi gas.

The blackbody pressure and energy are calculated as

$\label {Eqn:eos4a} P_{\rm rad} = {a T^4 \over 3}$
$\label {Eqn:eos4b} \epsilon_{\rm rad} = { 3 P_{\rm rad} \over \rho} \,$

where $$a$$ is related to the Stephan-Boltzmann constant $$\sigma_B \, = \, a c/4$$, and $$c$$ is the speed of light. The ion portion of each routine is the ideal gas of () with $$\gamma \, = \, 5/3$$. The number densities of free electrons $$N_{\rm ele}$$ and positrons $$N_{\rm pos}$$ in the noninteracting Fermi gas formalism are given by

$\label {Eqn:eos6a} N_{\rm ele} = {8 \pi \sqrt{2} \over h^3} \ m_{\rm e}^3 \ c^3 \ \beta^{3/2} \ \left[ F_{1/2}(\eta,\beta) \ + \ F_{3/2}(\eta,\beta) \right]$
$\label {Eqn:eos6b} N_{\rm pos} = {8 \pi \sqrt{2} \over h^3} \ m_{\rm e}^3 \ c^3 \ \beta^{3/2} \left[ F_{1/2} \left( -\eta - 2/\beta, \beta \right) \ + \ \beta \ F_{3/2} \left( -\eta - 2 /\beta, \beta \right) \right] \enskip ,$

where $$h$$ is Planck’s constant, $$m_{\rm e}$$ is the electron rest mass, $$\beta \: = \: k T / (m_{\rm e} c^2)$$ is the relativity parameter, $$\eta \: = \: \mu / k T$$ is the normalized chemical potential energy $$\mu$$ for electrons, and $$F_{k}(\eta,\beta)$$ is the Fermi-Dirac integral

$\label{Eqn:eos7} F_{k}(\eta,\beta) = \int\limits_{0}^{\infty} \ {x^{k} \ (1 + 0.5 \ \beta \ x)^{1/2} \ dx \over \exp(x - \eta) + 1 }\ .$

Because the electron rest mass is not included in the chemical potential, the positron chemical potential must have the form $$\eta_{{\rm pos}} \, = \, -\eta - 2/\beta$$. For complete ionization, the number density of free electrons in the matter is

$\label{Eqn:eos8} N_{\rm ele,matter} = {\bar{Z} \over \bar{A}} \ N_a \ \rho = \bar{Z} \ N_{\rm ion} \ ,$

and charge neutrality requires

$\label{Eqn:eos9} N_{\rm ele,matter} = N_{\rm ele} - N_{\rm pos} \ .$

Solving this equation with a standard one-dimensional root-finding algorithm determines $$\eta$$. Once $$\eta$$ is known, the Fermi-Dirac integrals can be evaluated, giving the pressure, specific thermal energy, and entropy due to the free electrons and positrons. From these, other thermodynamic quantities such as $$\gamma_1$$ and $$\gamma_4$$ are found. Full details of this formalism may be found in Fryxell et al. (2000) and references therein.

The above formalism requires many complex calculations to evaluate the thermodynamic quantities, and routines for these calculations typically are designed for accuracy and thermodynamic consistency at the expense of speed. The Helmholtz EOS in Flash-X provides a table of the Helmholtz free energy (hence the name) and makes use of a thermodynamically consistent interpolation scheme obviating the need to perform the complex calculations required of the above formalism during the course of a simulation. The interpolation scheme uses a bi-quintic Hermite interpolant resulting in an accurate EOS that performs reasonably well.

The Helmholtz free energy,

$\label {Eqn:eos13a} F = \epsilon - T \ S$
$\label {Eqn:eos13b} dF = -S \ dT + {P \over \rho^2} \ d\rho \enskip ,$

is the appropriate thermodynamic potential for use when the temperature and density are the natural thermodynamic variables. The free energy table distributed with Flash-X was produced from the Timmes EOS (Timmes & Arnett 1999). The Timmes EOS evaluates the Fermi-Dirac integrals [Eqn:eos7] and their partial derivatives with respect to $$\eta$$ and $$\beta$$ to machine precision with the efficient quadrature schemes of Aparicio (1998) and uses a Newton-Raphson iteration to obtain the chemical potential of [Eqn:eos9]. All partial derivatives of the pressure, entropy, and internal energy are formed analytically. Searches through the free energy table are avoided by computing hash indices from the values of any given $$(T,\rho \bar{Z}/\bar{A})$$ pair. No computationally expensive divisions are required in interpolating from the table; all of them can be computed and stored the first time the EOS routine is called.

We note that the Helmholtz free energy table is constructed for only the electron-positron plasma, and it is a 2-dimensional function of density and temperature, i.e. $$F(\rho,{\rm T})$$. It is made with $${\bar {\rm A}} \, = \, {\bar {\rm Z}} = 1$$ (pure hydrogen), with an electron fraction $$Y_{\rm e} \, = \, 1$$. One reason for not including contributions from photons and ions in the table is that these components of the Helmholtz EOS are very simple (), and one doesn’t need fancy table look-up schemes to evaluate simple analytical functions. A more important reason for only constructing an electron-positron EOS table with $$Y_{\rm e} \, = \, 1$$ is that the 2-dimensional table is valid for any composition. Separate planes for each $$Y_{\rm e}$$ are not necessary (or desirable), since simple multiplication by $$Y_{\rm e}$$ in the appropriate places gives the desired composition scaling. If photons and ions were included in the table, then this valuable composition independence would be lost, and a 3-dimensional table would be necessary.

The Helmholtz EOS has been subjected to considerable analysis and testing (Timmes & Swesty 2000), and particular care was taken to reduce the numerical error introduced by the thermodynamical models below the formal accuracy of the hydrodynamics algorithm (Fryxell, et al. 2000; Timmes & Swesty 2000). The physical limits of the Helmholtz EOS are $$10^{-10}\,<\,\rho\,<\,10^{11}~({\rm g~cm}^{-3})$$ and $$10^{4}\,<\,T\,<\,10^{11}$$ (K). As with the gamma-law EOS, the Helmholtz EOS provides both forward and backward relations. In the case of the forward relation ($$\rho, T$$, given along with the composition) the table lookup scheme and analytic formulae directly provide relevant thermodynamic quantities. In the case of the backward relation ($$\rho, \epsilon$$, and composition given), the routine performs a Newton-Rhaphson iteration to determine temperature. It is possible for the input variables to be changed in the iterative modes since the solution is not exact. The returned quantities are thermodynamically consistent.

## Usage

### Initialization

The initialization function of the Eos unit physics/Eos/Eos_init is fairly simple for the two ideal gas gamma law implementations included. It gathers the runtime parameters and the physical constants needed by the equation of state and stores them in the data module. The Helmholtz EOS physics/Eos/Eos_init routine is a little more complex. The Helmholtz EOS requires an input file helm_table.dat that contains the lookup table for the electron contributions. This table is currently stored in ASCII for portability purposes. When the table is first read in, a binary version called helm_table.bdat is created. This binary format can be used for faster subsequent restarts on the same machine but may not be portable across platforms. The Eos_init routine reads in the table data on processor 0 and broadcasts it to all other processors.

### Runtime Parameters

Runtime parameters for the Gamma unit require the user to set the thermodynamic properties for the single gas. Eos/gamma, Eos/eos_singleSpeciesA, Eos/eos_singleSpeciesZ set the ratio of specific heats and the nucleon and proton numbers for the gas. In contrast, the Multigamma implementation does not set runtime parameters to define properties of the multiple species. Instead, the simulation Config file indicates the requested species, for example helium and oxygen can be defined as

SPECIES HE4
SPECIES O16


The properties of the gases are initialized in the file Simulation/Simulation_initSpecies.F90, for example

subroutine Simulation_initSpecies()
use Multispecies_interface, ONLY : Multispecies_setProperty
implicit none
#include "Simulation.h"
#include "Multispecies.h"
call Multispecies_setProperty(HE4_SPEC, A, 4.)
call Multispecies_setProperty(HE4_SPEC, Z, 2.)
call Multispecies_setProperty(HE4_SPEC, GAMMA, 1.66666666667e0)
call Multispecies_setProperty(O16_SPEC, A, 16.0)
call Multispecies_setProperty(O16_SPEC, Z, 8.0)
call Multispecies_setProperty(O16_SPEC, GAMMA, 1.4)
end subroutine Simulation_initSpecies


For the Helmholtz equation of state, the table-lookup algorithm requires a given temperature and density. When temperature or internal energy are supplied as the input parameter, an iterative solution is found. Therefore, no matter what mode is selected for Helmholtz input, the best initial value of temperature should be provided to speed convergence of the iterations. The iterative solver is controlled by two runtime parameters Eos/eos_maxNewton and Eos/eos_tolerance which define the maximum number of iterations and convergence tolerance. An additional runtime parameter for Helmholtz, Eos/eos_coulumbMult, indicates whether or not to apply Coulomb corrections. In some regions of the $$\rho$$-$$T$$ plane, the approximations made in the Coulomb corrections may be invalid and result in negative pressures. When the parameter eos_coulombMult is set to zero, the Coulomb corrections are not applied.

### Direct and Wrapped Calls

The primary function in the Eos unit operates on a vector, taking density, composition, and either temperature, internal energy, or pressure as input, and returning $$\gamma_1$$, and either the pressure, temperature or internal energy (whichever was not used as input). This equation of state interface is useful for initializing a problem. The user is given direct control over the input and output, since everything is passed through the argument list. Also, the vector data format is more efficient than calling the equation of state routine directly on a point by point basis, since it permits pipelining and provides better cache performance. Certain optional quantities such electron pressure, degeneracy parameter, and thermodynamic derivatives can be calculated by the physics/Eos/Eos function if needed. These quantities are selected for computation based upon a logical mask array provided as an input argument. A .true. value in the mask array results in the corresponding quantity being computed and reported back to the calling function. Examples of calling the basic implementation Eos are provided in the API description, see physics/Eos/Eos.

The hydrodynamic and burning computations repeatedly call the Eos function to update pressure and temperature during the course of their calculation. Typically, values in all the cells of the block need of be updated in these calls. Since the primary Eos interface requires the data to be organized as a vector, using it directly could make the code in the calling unit very cumbersome and error prone. The wrapper interface, physics/Eos/Eos_wrapped provides a means by which the details of translating the data from block to vector and back are hidden from the calling unit. The wrapper interface permits the caller to define a section of block by giving the limiting indices along each dimension. The Eos_wrapped routine translates the block section thus described into the vector format of the physics/Eos/Eos interface, and upon return translates the vector format back to the block section. This wrapper routine cannot calculate the optional derivative quantities. If they are needed, call the Eos routine directly with the optional mask set to true and space allocated for the returned quantities.

## Unit Test

The unit test of the Eos function can exercise all three implementations. Because the Gamma law allows only one species, the setup required for the three implementations is specific. To invoke any three-dimensional Eos unit test, the command is:

./setup unitTest/Eos/implementation -auto -3d

where implementation is one of Gamma, Multigamma, Helmholtz. The Eos unit test works on the assumption that if the four physical variables in question (density, pressure, energy and temperature) are in thermal equilibrium with one another, then applying the equation of state to any two of them should leave the other two completely unchanged. Hence, if we initialize density and temperature with some arbitrary values, and apply the equation of state to them in MODE_DENS_TEMP, then we should get pressure and energy values that are thermodynamically consistent with density and temperature. Now after saving the original temperature value, we apply the equation of state to density and newly calculated pressure. The new value of the temperature should be identical to the saved original value. This verifies that the Eos unit is computing correctly in MODE_DENS_PRES mode. By repeating this process for the remaining two modes, we can say with great confidence that the Eos unit is functioning normally.

In our implementation of the Eos unit test, the initial conditions applied to the domain create a gradient for density along the $$x$$ axis and gradients for temperature and pressure along the $$y$$ axis. If the test is being run for the Multigamma or Helmholtz implementations, then the species are initialized to have gradients along the $$z$$ axis.