EOS_ybrid

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_POLYTROPE	REAL


eos_k	Scope: shared from EOS_2D_POLYTROPE	REAL


gamma_ini	Scope: shared from EOS_2D_POLYTROPE	REAL

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