Improved simulations of relativistic
stellar core collapse
José A. Font
Departamento de Astronomía y Astrofísica
Universidad de Valencia (Spain)
Collaborators:
• P. Cerdá-Durán, J.M. Ibáñez (UVEG)
• H. Dimmelmeier, F. Siebel, E. Müller (MPA)
• G. Faye (IAP), G. Schäfer (Jena)
• J. Novak (LUTH-Meudon)
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Outline of the talk
•
•
•
Numerical simulations of rotational stellar core collapse: gravitational waveforms
Relativistic hydrodynamics equations in conservation form (Godunov-type
schemes)
Approximations for the gravitational field equations (elliptic equations – finitedifference schemes, pseudo-spectral methods)
•
•
•
CFC (2D/3D)
•
CFC+ (2D)
Axisymmetric core collapse in characteristic numerical relativity
Improved means:
•
Treatment of gravity: from CFC to CFC+, and Bondi-Sachs
•
Modified CFC equations (high-density NS, BH formation)
•
Dimensionality: from 2D to 3D
•
Collapse dynamics: inclusion of magnetic fields
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Astrophysical motivation
General relativity and relativistic hydrodynamics play a major role in the description of
gravitational collapse leading to the formation of compact objects (neutron stars and black
holes): Core-collapse supernovae, black hole formation (and accretion), coalescing compact binaries
(NS/NS, BH/NS, BH/BH), gamma-ray bursts.
Time-dependent evolutions of fluid flow coupled to the spacetime geometry only possible through
accurate, large-scale numerical simulations. Some scenarios can be described in the test-fluid
approximation: hydrodynamical computations in curved backgrounds (highly mature nowadays).
(see e.g. Font 2003 online article: relativity.livingreviews.org/Articles/lrr-2003-4/index.html).
The (GR) hydrodynamic equations constitute a nonlinear hyperbolic system.
Solid mathematical foundations and accurate numerical methodology imported from CFD. A
“preferred” choice: high-resolution shock-capturing schemes written in conservation form.
The study of gravitational stellar collapse has traditionally been one of the primary problems in relativistic
astrophysics (for about 40 years now). It is a distinctive example of a research field in astrophysics where
essential progress has been accomplished through numerical modelling with gradually increasing levels of
complexity in the input physics/mathematics.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Introduction: supernova core collapse in a nutshell
The study of gravitational collapse of massive stars largely pursued numerically
over the years. Main motivation in May and White’s 1967 first one-dimensional
numerical relativity code.
Current standard model for a core collapse (type II/Ib/Ic) supernova: (from simulations!
[Wilson et al (late 1980s), MPA, Oak Ridge, University of Arizona (ongoing)])
• Nuclear burning in massive star yields shell structure. Iron core with 1.4 solar masses
and 1000 km radius develops in center. EoS: relativistic degenerate fermion gas, =4/3.
• Instability due to photo-disintegration and e- capture. Collapse to nuclear matter
densities in ~100ms.
• Stiffening of EoS, bounce, and formation of prompt shock.
• Stalled shock revived by neutrinos depositing energy behind it (Wilson 1985). Delayed
shock propagates out and disrupts envelope of star.
• Nucleosynthesis, explosion expands into interstellar matter. Proto-neutron star cools
and shrinks to neutron star.
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Introduction (continued)
May & White’s formulation and 1d code used by many groups to study core collapse.
Most investigations used artificial viscosity terms in the (Newtonian) hydro equations to handle
shock waves.
The use of HRSC schemes started in 1989 with the Newtonian simulations of Fryxell, Müller &
Arnett (Eulerian PPM code).
Relativistic simulations of core collapse with
HRSC schemes are still scarce.
Basic dynamics of the collapse at a glance:
1d core collapse simulations
Nonspherical core collapse simulations in GR
very important:
1. To produce and extract gravitational waves
consistently.
2. To explain rotation of newborn NS.
3. Collapse to NS is intrinsically relativistic
(2M/R ~0.2-0.4)
(let alone to BH!)
Romero et al 1996 (radial gauge polar slicing).
Purely hydrodynamical (prompt mechanism) explosion. No
microphysics or -transport included!
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Multidimensional core collapse & gravitational waves
Numerical simulations of stellar core collapse are nowadays highly motivated by the prospects
of direct detection of the gravitational waves (GWs) emitted.
GWs, ripples in spacetime generated by aspherical concentrations of accelerating matter, were predicted
by Einstein in his theory of general relativity. Their amplitude on Earth is so small (about 1/100th of the size
of an atomic nucleus!) that they remain elusive to direct detection (only indirectly “detected” in the
theoretical explanation of the orbital dynamics of the binary pulsar PSR 1913+16 by Hulse & Taylor (Nobel
laureates in physics in 1992).
International network of resonant bar detectors
International network of interferometer detectors
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Core collapse & gravitational waves (continued)
• GWs are dominated by a burst associated with the bounce. If rotation is present, the GWs
large amplitude oscillations associated with pulsations in the collapsed core (Mönchmeyer et al 1991;
Yamada & Sato 1991; Zwerger & Müller 1997; Rampp et al 1998 (3D!)).
• GWs from convection dominant on longer timescales (Müller et al 2004).
• Müller (1982): first numerical evidence of the low gravitational wave efficiency of the core
collapse scenario: E<10-6 Mc2 radiated as gravitational waves. (2D simulations, Newtonian,
finite-difference hydro code).
• Bonazzola & Marck (1993): first 3D simulations of the infall phase using pseudo-spectral
methods. Still, low amount of energy is radiated in gravitational waves, with little dependence
on the initial conditions.
• Zwerger & Müller (1997): general relativity counteracts the stabilizing effect of rotation. A
bounce caused by rotation will occur at larger densities than in the Newtonian case
 need for relativistic simulations:
Dimmelmeier et al 2001, 2002; Siebel et al 2003; Shibata & Sekiguchi 2004, 2005; Cerdá-Durán et al 2005.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Supernova codes vs core collapse numerical relativity codes
State-of-the art supernova codes are (mostly) based on Newtonian hydrodynamics (e.g. MPA
group, Oak Ridge National Laboratory group).
• Strong focus on microphysics (elaborate EoS, transport schemes for neutrinos – computationally
challenging).
• Often use of the most advanced initial models from stellar evolution.
• Simple treatment of gravity (Newtonian, possibly relativistic corrections).
However …
no generic explosions yet obtained! (even with most sophisticated multi-dimensional models)
Core collapse numerical relativity codes (mostly) originate from vacuum Einstein codes (e.g.
Whisky (EU), Shibata’s).
• No microphysics: matter often restricted to ideal fluid EoS.
• Simple initial (core collapse) models (uniformly or differentially rotating polytropes).
• Exact or approximate Einstein equations for spacetime metric (inherit the usual complications found in
numerical relativity: formulations of the field equations, gauge freedom, long-term numerical stability, etc).
Our approach: flux-conservative hyperbolic formulation for the hydrodynamics
CFC, CFC+, and Bondi-Sachs for the Einstein equations
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
3+1 General Relativistic Hydrodynamics equations (1)

  ( u )  0
T   0
p  p(  ,  )
Equations of motion:
[1]
[4]
local conservation laws of density current
(continuity equation) and stress-energy (Bianchi
identities)
[1]
Perfect fluid stress-energy tensor
T

 
 hu u  pg

Introducing an explicit coordinate chart:
Different formulations exist depending on:
1.
The choice of time-slicing: the level
surfaces of
can be spatial (3+1) or
0
null (characteristic) x
2.
The choice of physical (primitive)
variables (, , ui …)


(

g

u
)0

x




(

g
T
)


g

T

x 
Wilson (1972) wrote the system as a set of advection equation
within the 3+1 formalism. Non-conservative.
Conservative formulations well-adapted to numerical
methodology are more recent:
• Martí, Ibáñez & Miralles (1991): 1+1, general EOS
• Eulderink & Mellema (1995): covariant, perfect fluid
• Banyuls et al (1997): 3+1, general EOS
• Papadopoulos & Font (2000): covariant, general EOS
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
3+1 General Relativistic Hydrodynamics equations (2)
 ( u  )  0
T   0
p  p(  ,  )
[1]
[4]
[1]
1
R   g  R  8T [10]
2
Einstein’s equations
Foliate the spacetime with t=const
spatial hypersurfaces

St
ds 2  ( 2    i )dt 2  2  dxi dt   dxi dx
i
i
ij
n
t
Let n be the unit timelike 4-vector
orthogonal to St such that
Eulerian observers
v
n  i
nu
vi 
1 u
i 




t


 u

i
n
u: fluid’s 4-velocity, p: isotropic pressure,  : rest-mass density
 : specific internal energy density, e=( 1+ ): energy density
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
1

( t   i  i )
j
3+1 General Relativistic Hydrodynamics equations (3)
Replace the “primitive variables” in terms of the “conserved variables” :
 D  W


i
w   , v ,   S j  hW 2 v j

2
E


hW
p



W 2  1 /(1  v j v j )
h  1  
p

First-order flux-conservative hyperbolic system
 

1    u ( w) 


t
g 
where
i 
 g f ( w)   
  s ( w)
i

x

 
u ( w)  D, S j , E  D 
Banyuls et al, ApJ, 476,
221 (1997)
Font et al, PRD, 61,
044011 (2000)
is the vector of conserved variables
i
i 
  i i   i i 




i
i
i

, S j  v 
  p j , E  D v 
  pv 
f ( w)   D v 

  
 
 

 


   0  ln 
 
  gj

 0  

s ( w)   0, T 
  gj ,   T
 T   


x

 x
 

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fluxes
sources
Nonlinear hyperbolic systems of conservation laws (1)
For nonlinear hyperbolic systems classical
solutions do not exist in general even for
smooth initial data. Discontinuities develop
after a finite time.
For hyperbolic systems of conservation laws,
schemes written in conservation form
guarantee that the convergence (if it exists) is
to one of the weak solutions of the original
system of equations (Lax-Wendroff theorem
1960).
A scheme written in conservation form reads:
ˆ  n
 n 1  n t ˆ  n  n
n
n
n
u j  u j  ( f (u j r , u j  r 1 ,, u j  q )  f (u j r 1 , u j  r ,, u j  q 1 ))
x
ˆ  
 
̂

where f is a consistent numerical flux function: f (u , u ,, u )  f (u )
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Nonlinear hyperbolic systems of conservation laws (2)
The conservation form of the scheme is ensured by starting with the integral version of the PDE
in conservation form. By integrating the PDE within a spacetime computational cell
n n 1
the numerical flux function is an approximation
flux across the interface:
[ x j 1/ 2 , x j 1/ 2 ] to[tthe
, t time-averaged
]
ˆ
1 t n1  
f j 1/ 2   n f (u ( x j 1/ 2 , t )) dt
t t
Key idea: a possible
 procedure is to
calculate
solving
u ( x j1by
/ 2 , t)
Riemann problems at every cell
interface (Godunov)
The flux integral depends on the
 solution at the
numerical interfaces
the time step
u ( xduring
j1/ 2 , t )
When a Cauchy problem described by a set of
continuous PDEs is solved in a discretized form the
numerical solution is piecewise constant (collection of
local Riemann problems).

 n n
u ( x j 1/ 2 , t )  u (0; u j , u j 1 )
Riemann solution for the left and right
states along the ray x/t=0.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Nonlinear hyperbolic systems of conservation laws (3)
Any FD scheme must be able to handle discontinuities in a satisfactory way.
1.
1st order accurate schemes (Lax-Friedrich): Non-oscillatory but inaccurate across
discontinuities (excessive diffusion)
2.
(standard) 2nd order accurate schemes (Lax-Wendroff): Oscillatory across discontinuities
3.
2nd order accurate schemes with artificial viscosity
4.
Godunov-type schemes (upwind High Resolution Shock Capturing schemes)
Lax-Wendroff numerical solution of
Burger’s equation at t=0.2 (left) and
t=1.0 (right)
2nd order TVD numerical solution of
Burger’s equation at t=0.2 (left) and
t=1.0 (right)
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Nonlinear hyperbolic systems of conservation laws (4)
rarefaction wave
shock front
Solution at time n+1 of the two Riemann
problems at the cell boundaries xj+1/2 and xj-1/2
Spacetime evolution of the two Riemann
problems at the cell boundaries xj+1/2 and xj-1/2.
Each problem leads to a shock wave and a
rarefaction wave moving in opposite
directions
(Piecewise constant) Initial data at time n for the
two Riemann problems at the cell boundaries
xj+1/2 and xj-1/2
cell boundaries where fluxes are required
ˆ n 
 n 1  n t  ˆ n
uj  uj 
 f j 1/ 2  f j 1/ 2 

x 
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Approximate Riemann solvers
In Godunov’s method the structure of the Riemann solution is “lost” in the cell averaging process (1st order in
space).
The exact solution of a Riemann problem is computationally expensive, particularly in multidimensions and for
complicated EoS.
Relativistic multidimensional problems: coupling of all flow velocity components through the Lorentz factor.
• Shocks: increase in the number of algebraic jump (RH) conditions.
• Rarefactions: solving a system of ODEs.
This motivated the development of approximate
(linearized) Riemann solvers.
Roe-type SRRS (Martí et al 1991; Font et al 1994)
Based on the exact solution of Riemann problems
corresponding to a new system of equations
obtained by a suitable linearization of the original
one. The spectral decomposition of the Jacobian
matrices is on the basis of all solvers.
Exact SRRS (Martí & Müller 1994; Pons et al 2000)
Approach followed by an important subset of shockcapturing schemes, the so-called Godunov-type
methods (Harten & Lax 1983; Einfeldt 1988).
HLLE SRRS (Schneider et al 1993)
Two-shock approximation (Balsara 1994)
ENO SRRS (Dolezal & Wong 1995)
Roe GRRS (Eulderink & Mellema 1995)
Upwind SRRS (Falle & Komissarov 1996)
Glimm SRRS (Wen et al 1997)
Iterative SRRS (Dai & Woodward 1997)
Marquina’s FF (Donat et al 1998)
Martí & Müller, 2003
Living Reviews in Relativity
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www.livingreviews.org
A standard implementation of a HRSC scheme
1. Time update: Conservation form algorithm
ˆ n 
 n1  n t  ˆ n
u j  u j   f j 1/ 2  f j 1/ 2 

x 
In practice: 2nd or 3rd order time accurate,
conservative Runge-Kutta schemes (Shu & Osher
1989)
3. Numerical fluxes: Approximate Riemann
solvers (Roe, HLLE, Marquina). Explicit use
of the spectral information of the system
2. Cell reconstruction: Piecewise constant
(Godunov), linear (MUSCL, MC, van Leer),
parabolic (PPM, Colella & Woodward 1984)
interpolation procedures of state-vector variables
from cell centers to cell interfaces.
5
ˆ 1  

~
~
f i   f i ( wR )  f i ( wL )   n ~n Rn 
2
n 1

5
~
U( wR )  U( wL )   ~n Rn
n 1
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HRSC schemes: numerical assessment
• Stable and sharp discrete shock
profiles
Relativistic shock reflection
Shock tube test
• Accurate propagation speed of
discontinuities
• Accurate resolution of multiple
nonlinear structures: discontinuities,
raraefaction waves, vortices, etc
V=0.99999c (W=224)
Simulation of a extragalactic relativistic jet
Wind accretion onto a Kerr black hole (a=0.999M)
Scheck et al, MNRAS, 331, 615 (2002)
Font et al, MNRAS, 305, 920 (1999)
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Relativistic Rotational Core Collapse (CFC)
Dimmelmeier, Font & Müller, ApJ, 560, L163 (2001); A&A, 388, 917 (2002a); A&A, 393, 523 (2002b)
Goals
extend to GR previous results on Newtonian rotational core collapse (Zwerger & Müller 1997)
determine the importance of relativistic effects on the collapse dynamics (angular
momentum)
compute the associated gravitational radiation (waveforms)
Model assumptions
axisymmetry and equatorial plane symmetry
(uniformly or differentially) rotating 4/3 polytropes in equilibrium as initial models (Komatsu,
Eriguchi & Hachisu 1989). Central density 1010 g cm-3 and radius 1500 km. Various rotation
profiles and rotation rates
simplified EoS: P = Ppoly + Pth (neglect complicated microphysics and allows proper treatment
of shocks)
constrained system of the Einstein equations (IWM conformally flat condition)
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CFC metric equations
 ijCFC   4 ij
In the CFC approximation (Isenberg 1985; Wilson & Mathews 1996) the ADM 3+1 equations
 t ij  2K ij   i  j   j  i


 t K ij  i  j   Rij  KKij  2 K im K mj   m m K ij  K im j  m  K jm i  m  8T ij
R  K 2  K ij K ij  16 2T 00  0


 i K ij   ij K  8S j  0
reduce to a system of five coupled, nonlinear elliptic equations for the lapse function,
conformal factor, and the shift vector:
ij


K
K
ij
5
2

  2 W  P 

16 

ij


7
K
K
ij
5
2

  2 h 3W  2  5P 

16 



  1
 i  164 S i  2 K ij j  6    i  k  k
  3
Solver 1: Newton-Raphson iteration. Discretize
equations and define root-finding strategy.
Solver 2: Conventional integral Poisson
iteration. Exploits Poisson-like structure of metric
equations, uk=S(ul). Keep r.h.s. fixed, obtain linear
Poisson equations, solve associated integrals, then
iterate until nonlinear equations converge.
Both solvers feasible in axisymmetry but no
extension to 3D possible.
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Animation of a representative rotating core collapse simulation
For movies of additional models visit:
www.mpa-garching.mpg.de/rel_hydro/axi_core_collapse/movies.shtml
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Central Density Gravitational Waveform
HRSC scheme:
Larger central densities in
relativistic models
Similar gravitational radiation
amplitudes (or smaller in the
GR case)
GR effects do not improve the
chances for detection (at least
in axisymmetry)
Type II “multiple bounce”
Dashed line: Newtonian
“transition”
Solid line: relativistic simulation
Type I “regular”
PPM + Marquina flux-formula
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Gravitational Wave Signals
www.mpa-garching.mpg.de/Hydro/RGRAV/index.html
Influence of relativistic effects on signals: Investigate amplitude-frequency diagram
Spread of the 26 models does not change much
Signal of a galactic supernova detectable
On average: Amplitude → Frequency ↑
If close to detection threshold: Signal could fall out of the sensitivity window!
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CFC+ metric equations
Cerdá-Duran, Faye, Dimmelmeier, Font, Ibáñez, Müller, and Schäfer, A&A, in press (2005)
CFC+ metric:
 ijCFC    ijCFC  hijTT ,
trhTT  0
(ADM gauge)
The second post-Newtonian deviation from isotropy is the solution of:
hijTT 
1 TTkl
1

(

16

v
v

4

U

U
)


 6
k l
k
l
4 ij
c
c 
(complicated) transverse, traceless projection operator
(Schäfer 1990)
Newtonian potential
Modified equations for , i and  (with respect to CFC):

K ij K ij 
2

  2  hW  P 

16 

5
ij

 1 TT
7
K
K
ij
5
2
  hij  ijU
  2 h 3W  2  5 P 

 c2
16





  1
 i  164 S i  2 K ij j  6    i  k  k
  3
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CFC+ metric equations (2)
TT
We can solve the
by introducing some
hequations
ij
intermediate potentials:
1
7
1
Sij   ij S kk  3 x k  (i S j ) k  3 (i S j )  x j  i S kk
2
4
4
1
1
1
1
1
  iT j  x k  ij S k  x k x l  ij S kl   ij S   i R j
4
2
4
4
4
hijTT 



1
M2 i
1
i
k
i
k
3
S    v vk x  x U  x  kU d x 
n  O 2 
r
2r
r 
i

1  i j  ij
1
S     v v 
 x j  iU d 3 x  O 2 
r
2
r 


S i  (4vi v j   iU jU ) x j
S ij  4vi v j   iU jU
S
 4vi v j x i x j
T i
R i
 (4v j v j   jU jU ) xi
  i  kU lU x k x l
16 elliptic linear equations
Linear solver: LU decomposition
using standard LAPACK routines
ij
S 
1
1
k l 3

v
v
x
x
d
x

O
 2
k l
r
r 
Ti 
1
M i
1
k i
i
3

v
v
x

x
U
d
x

n

O
 2
k
r
2r
r 


Boundary conditions
2
Multipole development in compactsupported integrals
M2 i
1
R 
n  O 2 
r
r 
i
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
CFC+ results: rotating neutron stars
Initial models (KEH method)
Model
Axis ratio
/K
Mass
(sun)
RNS0
1.00
0.00
1.40
RNS1
0.95
0.42
1.44
RNS2
0.85
0.70
1.51
RNS3
0.75
0.87
1.59
RNS4
0.70
0.93
1.63
RNS5
0.65
0.98
1.67
Study the time-evolution of
equilibrium models under the
effect of a small amplitude
perturbation.
Computation of radial and quasiradial mode-frequencies (code
validation: comparison between
CFC and CFC+ results, and with
those of an independent full GR
code)
Equatorial profiles of the non-vanishing components of hij for the
sequence of rigidly rotating models RNS0 to RNS5
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
rotating NS
spherical NS
CFC+:radial modes of spherical NS
quasi-radial modes of rotating NS
No noticeable differences between CFC and CFC+
Good agreement in the mode frequencies (better than 2%), also with
results from a full GR 3D code (Font et al 2002)
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
CFC+: core collapse dynamics (1)
Type I (regular collapse)
Type III (rapid collapse)
Relative differences between CFC and CFC+ for the central density and the lapse remain of
the order of 10-4 or smaller throughout the collapse and bounce.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
CFC+: core collapse dynamics (2)
Type II (multiple bounce)
Extreme case (torus-like structure)
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
CFC+: core collapse waveforms
Two distinct ways to extract
waveforms:
From the quadrupole formula:

AE 2 t  r / c 
1 15
hquad x, t  
sin  2 20
8 
r
From the metric hij:

AE2 t  r / c 
1 15
h2 PN x , t  r / c  
sin  2 20
64 
r

1
 hquad x , t 
8
Offset correction (dashed line)
h2PN-corrected  h2PN  a ij hijTT
Absolute differences
between CFC and CFC+
waveforms.
No significant differences
found.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
“Mariage des Maillages”
HRSC schemes for hydrodynamics and spectral methods for metric
Reference: Dimmelmeier, Novak, Font, Müller, Ibáñez, PRD 71, 064023 (2005)
The extension of our code to 3D has been possible thanks to the use of a metric solver based on integral
Poisson iteration (as solver 2) but using spectral methods.
MdM idea: Use spectral methods for the metric (smooth functions, no discontinuities) and HRSC schemes for
the hydro (discontinuous functions).
Valencia/Meudon/Garching collaboration. New metric solver uses publicly available package in C++ from
Meudon group (LORENE). Communication between finite-difference grid and spectral grid necessary (highorder interpolators). It works!
Spectral solver uses several (3-6) radial domains (easy with
LORENE package):
• Nucleus limited by rd (domain radius parameter)
roughly at largest density gradient.
• Several shells up to rfd.
• Compactified radial vacuum domain out to spatial
infinity.
In contraction phase of core collapse, inner domain
boundaries are allowed to move (controlled by mass fraction
or sonic point).
The relation between the FD grid and the spectral grid
changes dynamically due to moving domains.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
MdM code: Importance of a moving spectral grid
In core collapse relevant radial scale contracts by a factor 100.
Spectral grid setup with moving domains allows to put resolution where needed.
Example: influence of bad spectral grid setup on collapse dynamics.
1. Domain radius rd must follow
contraction.
rd held fixed (10% initial rse)
wrong result!
2. Domain radius rd should stay fixed
at roughly rpns after core bounce.
final rd too large
bounce time
only 3 radial domains
3. More than 3 domains needed in
dynamical core collapse.
4. Compare with previous solvers in
axisymmetry: 33 collocation points
per domain sufficient.
Gibbs-type oscillations
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
MdM code: Oscillations of rotating neutron stars
Another stringent test: can code keep rotating neutron stars in equilibrium?
Test criterion: preservation of rotation velocity profile (here shown after 10 ms).
Compactified grid essential if rfd close to rstar (profile deteriorates only negligibly)
Axisymmetric oscillations in rotating neutron stars
can be evolved as in other codes.
No important differences between running the
code in 2d or 3d modes.
3d low resolution
2d high resolution
3d low resolution without artificial perturbation
Proof of principle: code is ready for simulations of
dynamical triaxial instabilities.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
MdM code: Generic nonaxisymmetric configurations
Explore nonaxisymmetric configurations in 3d.
Extension from axisymmetry to 3d trivial with LORENE.
Even in axisymmetry spectral solver uses  coordinate with 4 collocation points (shift vector
Poisson equation is calculated for Cartesian components).
Setup: rotating NS with strong (unphysical) nonaxisymmetric “bar” perturbation.
Rotation generates spiral arms
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
MdM code: Comparison with full GR core collapse
simulations by Shibata and Sekiguchi
Full GR simulations of axisymmetric core collapse
available recently (Shibata & Sekiguchi 2004).
Comparison between CFC and full GR possible!
Shibata & Sekiguchi used rotational core collapse
models with parameters close (but not equal) to the
ones used by Dimmelmeier, Font & Müller (2002).
Disagreements in the GW amplitude of about 20% at
the peak (core bounce) and up to a factor 2 in the
ringdown.
A3B2G4 (DFM 02)

W6 (20% gain at bounce!)
Most plausible reason for discrepancy: different
functional form of the density used in the wave
extraction method (W6) and the formulation (stress
formulation vs first moment of momentum density
formulation).
A3B2G4
Shibata & Sekiguchi
A3B2G4 (A/rse=0.32)
Shibata & Sekiguchi
(A/rse=0.25)
The qualitative difference found by Shibata &
Sekiguchi (2004) is due to the differences in the
collapse initial model, notably the small decrease of
the differential rotation length scale in their model.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
CFC metric equations:
modification to allow for black hole formation
Original CFC equations

K ij K
  2  W 2  P 

16

5
ij
It turns out to be essential to rescale some
of the hydro quantities with the appropriate
power of the conformal factor for the elliptic
solvers to converge to the correct solution:





7 K ij K ij 
2

  2  h 3W  2  5P 

16 

5


*   6
S i*   6 S i
  1
 i  164 S i  2 K ij j  6    i  k  k
  3
*
 * , S i* , P
one
To obtain
needs to first compute the
conformal factor, which is
obtained from the evolution
equation
 
 k  k
t 6
P*   6 P
 1 
K ij K ij 
2

5

  2    hW  P  

16 



 1 
7 K ij K ij 
2

5

   2   h 3W  5  5P  

16 

 


  1
 i  16 2 S i  2 K ij j  6    i  k  k
  3
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Black hole formation (spherical symmetry)
with rescaling
without rescaling
High central density TOV solution
Collapse of a (perturbed) unstable neutron star to
a black hole in spherical symmetry.
Collapse can be followed well beyond formation of
an apparent horizon.
Central density grows by 6 orders of magnitude,
central lapse function drops to 0.0002.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Rotational core collapse to high-density NS: CFC vs CFC+
Model M7C5 (Shibata and Sekiguchi 2005):
• Differential rotation parameter A/R=0.1
• Baryon rest mass M*=2.464
• Angular momentum J/M2=0.664
• Polytropic EOS (=4/3, k=7x1014 (cgs))
Excellent agreement with the full BSSN simulations
of Shibata & Sekiguchi (2005)
max=1.4x1015
min=0.42
GW amplitude larger at
bounce with CFC+
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Core collapse simulations using the Einstein
for the Bondi metric
equations
Reference: Siebel, Font, Müller, and Papadopoulos, PRD 67, 124018 (2003)
V

ds 2   e 2   U 2 r 2 e 2 du 2  2e 2  dudr  2Ur 2 e 2 dud  r 2 e 2 d 2  e  2 sin 2 d 2 
r

The metric functions V ,U ,  and  only depend on the coordinate s u, r and 
 

1
1

Gab  Rab  g ab R
Rab    h ua ub  g ab   pg ab  Ricci tensor
2
2

 

r
r
2
Rrr   ,r   ,r 
4
2

2r 2 Rr  r 4 e 2 (   )U ,r

,r
2


 2r 2   ,r   ,r  2 ,r ,   ,  2 ,r cot  
r


1
 r 2 e 2  g AB RAB  2V,r  r 4 e 2 (   ) (U ,r ) 2  r 2U ,r  4rU ,  r 2U ,r cot 
2
 4rU cot   2e 2 (   )  1  (3 ,   , ) cot    ,   ,  (  , ) 2  2 , ( ,   , )

 r 2 e 2  g  R  2r (r ) ,ur  (1  r ,r )V,r  (r ,rr   ,r )V  r (1  r ,r )U ,
 r 2 (cot    , )U ,r  e 2 (   )  1  (3 ,  2  , ) cot    ,  2 , ( ,   , )
 rU 2r ,r  2 ,  r ,r cot   3 cot  
Hypersurface equations: hierarchical set for
Evolution equation for
(r ),ur
 ,r ,U ,r and V,r
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005

The light-cone problem is formulated in the region of spacetime between a
timelike worldtube at the origin of the radial coordinate and future null infinity.
Initial data for  are prescribed on an initial light cone u=0.
Boundary data for , U, V and  are also required on the worldtube.
For the general relativistic hydrodynamics equations we use a covariant (form
invariant respect to the spacetime foliation) formulation developed by Papadopoulos
and Font (PRD, 61, 024015 (2000)) which casts the equations in flux-conservative,
first-order form.
Gravitational waves at null infinity:
Null code test: time of bounce
• Bondi news function (from the metric variables expansion
at scri)
• Approximate gravitational waves (Winicour 1983, 1984,
1987):
150 x
1 x4
• Quadrupole news
Grid 1 : r 
• First moment of momentum formula
 
Grid 2 : r  100 tan  x 
2 
Time of bounce: 39.45 ms (null code 1), 38.32 ms (CFC
code), 38.92 ms (null code 2).
Good agreement between independent codes (less than
1% difference).
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Gravitational waves: consistency & disagreement
Good agreement in the computation of the GW strain using the
quadrupole moment and the first moment of momentum formula.
Equivalence valid in the Minkowskian limit and for small
velocities, which explains the small differences.
But Siebel et al (2002)
found excellent
agreement between the
quadrupole news and the
Bondi news when
calculating GWs from
pulsating relativistic stars.
Quadrupole news
rescaled by a factor
50.
A possible explanation: different velocities involved in both scenarios, 10-510-4c for a pulsating NS and 0.2c in core collapse.
Functional form for the quadrupole moment established in the slow motion
limit on the light cone may not be valid.
Large disagreement between Bondi news and quadrupole news, both in amplitude and frequency of the
signal.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
3+1 General Relativistic (Ideal)
Magnetohydrodynamics equations (1)
GRMHD: Dynamics of relativistic, electrically conducting fluids in the presence of magnetic
fields.
Ideal GRMHD: Absence of viscosity effects and heat conduction in the limit of infinite
conductivity (perfect conductor fluid).
The stress-energy tensor includes the contribution from the perfect fluid and from the magnetic
 observer comoving with the fluid.
field
measured by b
the
T   hu  u  p g   b  b

T   TPF  TEM
TPF  hu  u  pg 

TEM
1
1


 F  F  g  F  F   u  u  g  b 2  b  b
4
2


F     u b
Ideal MHD condition:
electric four-corrent must
be finite.

F  u  0
J  qu   F  u
  
with the definitions:
b 2  b b
b2
p  p
2
b2

h  h

Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005

3+1 General Relativistic (Ideal)
Magnetohydrodynamics equations (2)
  ( u  )  0
1.
Conservation of mass:
2.
Conservation of energy and momentum:
3.
Maxwell’s equations:
   F 
  T   0
1
 0 , F   u  B  u B  
W  
 
1 
 t
•
Induction equation:
•
Divergence-free constraint:
  B    v    B
 
 B  0
i

 u
 g f  
1 

s

i

x
g 
 t

Adding all up: firstorder, fluxconservative,
hyperbolic system of
balance laws
+ constraint
(divergence-free
condition)
  Bi
0
i
x
0


i
~


D


Dv
gj




 


 T     gj  

2
i

i
i
~

S
  
h W v j v  p  j  b b j

  
 x
 
u  j  f i   2 i
s

   0  ln 

~  p   i /   Dv~ i  b 0b i 

 0 

h
W
v
 T   
 k


  T

B 
i
k
k
i


x

 
v~ B  v~ B
 


k


0


Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Solution procedure of the GRMHD equations
i


 g f  
1  u
s

i


x
 g  t


 Bi
x
i
0
Constrained transport scheme (Evans & Hawley 1988,
• Same HRSC schemes as for GRHD equations
Tóth 2000).
(HLL, Kurganov-Tadmor, Roe-type)
• Wave structure information obtained
• Primitive variable recovery more involved
Field components defined at cell interfaces. Zonecentered vector (needed for primitive recovery and cell
reconstruction & Riemann problem) obtained from
staggered field components:
Details: Antón, Zanotti, Miralles, Martí, Ibáñez, Font & Pons,
in preparation (2005)
Bi ,xj 
Update of field components:
B 
 Bix, j


B 
 Biy, j


x n 1
i, j
y n 1
i, j
n
n
 
t
i , j 1  i , j 
y
t
i 1, j  i , j 

x

1
Bix, j  Bix1, j
2








1 ˆy x
f Bi 1, j  fˆ y Bix, j  fˆ x Biy, j 1  fˆ x Biy, j
4
 
These equations conserve the discretization of
 B
i , j 

 
 B

i, j

Bix1, j  Bix, j
xi , j

Biy, j 1  Biy, j
yi , j
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005

GRMHD equations: code tests (1)
1D Relativistic Brio-Wu shock tube test (van Putten 1993, Balsara 2001)
Dashed line: wave structure in Minkowski spacetime at time t=0.4
Open circles: nonvanishing lapse function (2), at time t=0.2
Open squares: nonvanishing shift vector (0.4), at time t=0.16
HLL solver
1600 zones
CFL 0.5
Agreement with previous authors (Balsara 2001) regarding wave locations, maximum Lorentz factor
achieved, and numerical smearing of the solution.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
GRMHD equations: code tests (2)
density
Magnetized spherical accretion onto a Schwarzschild BH
Test difficulty: keeping the stationarity of the solution
Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003)

internal energy

radial velocity
radial magnetic field


Initial data: Magnetic field of the type
b  bon, btop
,0,of
0 the
hydrodynamic (Michel) accretion solution.
Radial magnetic
field component chosen to satisfy divergence-free condition, and its
strength is parametrized by the ratio:
2p
b2
HLL solver
100 zones
=1
Solid lines: analytic solution
Circles: numerical solution
(t=350M)
Increasing the grid resolution
shows that code is second-order
convergent irrespective of the
value of 
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
t
r
GRMHD equations: code tests (3)
Magnetized equatorial Kerr accretion (Takahashi
density
et al 1990, Gammie 1999)
Test difficulty: keeping the stationarity of the solution (algebraic
complexity augmented, Kerr metric)
Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003)
azimuthal
velocity
radial magnetic
field
Inflow solution determined by specifying 4 conserved quantities: the
mass flux FM, the angular momentum flux FL, the energy flux FE, and the
component F of the electromagnetic tensor.
a=0.5
FM=-1.0
FL=-2.815344
FE=-0.908382
F=0.5
HLL solver
azimuthal
magnetic field
Solid lines: analytic
solution
Circles: numerical
solution (t=200M)
second order convergence
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
GRMHD: spherical core collapse simulation
As a first step towards relativistic magnetized core collapse simulations we employ the test (passive) field
approximation for weak magnetic field.
• magnetic field attached to the fluid (does not backreact into the Euler-Einstein equations).
• eigenvalues (fluid + magnetic field) reduce to the fluid eigenvalues only.
HLL solver + PPM, Flux-CT, 200x10 zones
The divergence-free condition is fulfilled to good
precision during the simulation.
The amplification factor of the initial magnetic field
during the collapse is 1370.
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005
Summary of the talk
• Multidimensional simulations of relativistic core collapse feasible nowadays with current
formulations of hydrodynamics and Einstein’s equations.
• Results from CFC and CFC+ relativistic simulations of rotational core collapse to NS in
axisymmetry. Comparisons with full GR simulations show that CFC is a sufficiently accurate
approach.
• Modification of the original CFC equations to allow for collapse to high density NS and BH
formation.
• Ongoing work towards extending the SQF for GW extraction (1PN quadrupole formula).
• Axisymmetric core collapse simulations using characteristic numerical relativity show
important disagreement in the gravitational waveforms between the Bondi news and the
quadrupole news.
• 3d extension of the CFC core collapse code through the MdM approach (HRSC schemes for
the hydro and spectral methods for the spacetime).
• First steps towards GRMHD core collapse simulations (ongoing work)
Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005