EOS_Hybrid

Harry Dimmelmeier, Ian Hawke, Christian Ott

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Abstract

EOS_Hybrid. The equation of state used for “simple” core collapse simulations.

1 The equations

This equation provides the hybrid polytropic / ideal gas equation of state used by Dimmelmeier et al. [1] for supernova collapse. A thorn wanting to use this needs to use the CactusEOS interface found in EOS_Base.

The equations are

\begin {eqnarray} \label {eq:eosformulas} P & = & P_{\text {poly}} + P_{\text {th}} \\ \frac {\partial P}{\partial \rho } & = & \frac {\partial P_{\text {poly}}}{\partial \rho } + \frac {\partial P_{\text {th}_1}}{\partial \rho } + \frac {\partial P_{\text {th}_2}}{\partial \rho } \\ \frac {\partial P}{\partial \epsilon } & = & (\gamma _{\text {th}} - 1) \rho , \end {eqnarray}

where

\begin {eqnarray} \label {eq:eostemps} P_{\text {poly}} & = & K \rho ^{\gamma } \\ P_{\text {th}} & = & -K \frac {\gamma _{\text {th}} - 1}{\gamma - 1} \rho ^{\gamma } + (\gamma _{\text {th}} - 1) \rho \epsilon - (\gamma _{\text {th}} - 1) \frac {\gamma - \gamma _1}{\gamma _2 - 1} K \rho _{\text {nuc}}^{\gamma _1 - 1} \rho \\ \frac {\partial P_{\text {poly}}}{\partial \rho } & = & \gamma K \rho ^{\gamma - 1} \\ \frac {\partial P_{\text {th}_1}}{\partial \rho } & = & - \gamma K \frac {\gamma _{\text {th}} - 1}{\gamma - 1} \rho ^{\gamma -1} \\ \frac {\partial P_{\text {th}_2}}{\partial \rho } & = & (\gamma - 1) \epsilon - (\gamma _{\text {th}} - 1) \frac {\gamma - \gamma _1}{(\gamma _1 - 1)(\gamma _2 - 1)} K \rho _{\text {nuc}} (\gamma _1 - 1). \end {eqnarray}

These expressions do not include the conversions between cgs and Cactus units that are necessary inside the code.

For more details, a recent paper to start with would be [1].

References

[1]   H. Dimmelmeier, J. Novak, A. Font, J. M. Ibanez, E. Mueller Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations Phys. Rev. D71 064023 (2005)

2 Parameters

eos_gamma_supernuclear
 Scope: restricted REAL
Description: Gamma_2 for EOS
Range Default: 1.66666666666666
:

eos_gamma_th
 Scope: restricted REAL
Description: Gamma_th for EOS
Range Default: 1.5
:

rho_nuc
 Scope: restricted REAL
Description: Nuclear matter density
Range Default: 1.e-10
:

eos_gamma
 Scope: shared from EOS_2D_POLYTROPEREAL

eos_k
 Scope: shared from EOS_2D_POLYTROPEREAL

gamma_ini
 Scope: shared from EOS_2D_POLYTROPEREAL

3 Interfaces

General

Implements:

eos_hybrid

Inherits:

hydrobase

eos_2d_polytrope

Grid Variables

3.0.1 PRIVATE GROUPS

  Group Names     Variable Names   Details    
pressure_contributions compact 0
pressure_poly description Polytropic and thermal contributions to the pressure
pressure_th dimensions 3
distribution DEFAULT
group type GF
tags Prolongation=”None”
timelevels 1
variable type REAL

4 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinEOS/EOS_Hybrid. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

NONE

Scheduled Functions

CCTK_STARTUP

  eos_hybrid_startup

  setup the eos used by dimmelmeier et al. in core collapse

 

  After: eos_polytrope_startup
  Language: fortran
  Type: function

CCTK_ANALYSIS

  check_poly_contrib

  output the polytropic and thermal contributions

 

  Language: fortran
  Storage: pressure_contributions
  Triggers: pressure_contributions
  Type: function