GR-hydro

1
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)

2
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 finite-difference
  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

3
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/i
  ndex.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.

4
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 Whites 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.

5
Romero et al 1996 (radial gauge polar slicing).
Purely hydrodynamical (prompt mechanism)
explosion. No microphysics or ?-transport
included!
Introduction (continued)
May Whites 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).
Basic dynamics of the collapse at a glance 1d
core collapse simulations
Relativistic simulations of core collapse with
HRSC schemes are still scarce.

• Nonspherical core collapse simulations in GR very
  important
• To produce and extract gravitational waves
  consistently.
• To explain rotation of newborn NS.
• Collapse to NS is intrinsically relativistic
  (2M/R 0.2-0.4) (let alone to BH!)

6
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 191316 by Hulse Taylor (Nobel
laureates in physics in 1992).
International network of resonant bar detectors
International network of
interferometer detectors
7
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 Elt10-6 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.

8
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), Shibatas).
• 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
9
31 General Relativistic Hydrodynamics equations
(1)
Equations of motion local conservation laws of
density current (continuity equation) and
stress-energy (Bianchi identities)
Perfect fluid stress-energy tensor
Introducing an explicit coordinate chart

• Wilson (1972) wrote the system as a set of
  advection equation within the 31 formalism.
  Non-conservative.
• Conservative formulations well-adapted to
  numerical methodology are more recent
• Martí, Ibáñez Miralles (1991) 11, general
  EOS
• Eulderink Mellema (1995) covariant, perfect
  fluid
• Banyuls et al (1997) 31, general EOS
• Papadopoulos Font (2000) covariant, general
  EOS

• Different formulations exist depending on
• The choice of time-slicing the level surfaces of
  can be spatial (31) or null
  (characteristic)
• The choice of physical (primitive) variables (?,
  ?, ui )

10
31 General Relativistic Hydrodynamics equations
(2)
Einsteins equations
Foliate the spacetime with tconst spatial
hypersurfaces St
?
?n
?t
Let n be the unit timelike 4-vector orthogonal to
St such that
Eulerian observers
u fluids 4-velocity, p isotropic pressure, r
rest-mass density e
specific internal energy density, er( 1e )
energy density
11
31 General Relativistic Hydrodynamics equations
(3)
Replace the primitive variables in terms of the
conserved variables
First-order flux-conservative hyperbolic system
Banyuls et al, ApJ, 476, 221 (1997) Font et al,
PRD, 61, 044011 (2000)
where
is the vector of conserved variables
fluxes
sources
12
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
where is a consistent numerical flux
function
13
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
the numerical flux function
is an approximation to the time-averaged flux
across the interface
The flux integral depends on the solution at the
numerical interfaces during
the time step
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).
Key idea a possible procedure is to calculate
by solving Riemann problems at
every cell interface (Godunov)
Riemann solution for the left and right states
along the ray x/t0.
14
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 Burgers
equation at t0.2 (left) and t1.0 (right)
2nd order TVD numerical solution of Burgers
equation at t0.2 (left) and t1.0 (right)
15
Nonlinear hyperbolic systems of conservation laws
(4)
rarefaction wave
shock front
Solution at time n1 of the two Riemann problems
at the cell boundaries xj1/2 and xj-1/2
Spacetime evolution of the two Riemann problems
at the cell boundaries xj1/2 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
xj1/2 and xj-1/2
cell boundaries where fluxes are required
16
Approximate Riemann solvers

• In Godunovs 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.

Roe-type SRRS (Martí et al 1991 Font et al
1994) HLLE SRRS (Schneider et al 1993) Exact
SRRS (Martí Müller 1994 Pons et al
2000) 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) Marquinas FF (Donat
et al 1998)
This motivated the development of approximate
(linearized) Riemann solvers. 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. Approach followed by an
important subset of shock-capturing schemes, the
so-called Godunov-type methods (Harten Lax
1983 Einfeldt 1988).
Martí Müller, 2003 Living Reviews in Relativity
www.livingreviews.org
17
A standard implementation of a HRSC scheme
1. Time update Conservation form algorithm
In practice 2nd or 3rd order time accurate,
conservative Runge-Kutta schemes (Shu Osher
1989)
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.
3. Numerical fluxes Approximate Riemann solvers
(Roe, HLLE, Marquina). Explicit use of the
spectral information of the system
18
HRSC schemes numerical assessment
Shock tube test
Relativistic shock reflection

• Stable and sharp discrete shock profiles
• Accurate propagation speed of discontinuities
• Accurate resolution of multiple nonlinear
  structures discontinuities, raraefaction waves,
  vortices, etc

V0.99999c (W224)
Wind accretion onto a Kerr black hole
(a0.999M) Font et al, MNRAS, 305, 920 (1999)
Simulation of a extragalactic relativistic
jet Scheck et al, MNRAS, 331, 615 (2002)
19
Relativistic Rotational Core Collapse (CFC)
Dimmelmeier, Font Müller, ApJ, 560, L163
(2001) AA, 388, 917 (2002a) AA, 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)
20
CFC metric equations
In the CFC approximation (Isenberg 1985 Wilson
Mathews 1996) the ADM 31 equations reduce
to a system of five coupled, nonlinear elliptic
equations for the lapse function, conformal
factor, and the shift vector
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, ?ukS(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.
21
Animation of a representative rotating core
collapse simulation For movies of
additional models visit
www.mpa-garching.mpg.de/rel_hydro/axi_core_collaps
e/movies.shtml
22
Central Density Gravitational Waveform
HRSC scheme PPM Marquina flux-formula Solid
line relativistic simulation Dashed line
Newtonian 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)
transition Type II multiple
bounce Type I regular
23
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!
24
CFC metric equations
Cerdá-Duran, Faye, Dimmelmeier, Font, Ibáñez,
Müller, and Schäfer, AA, in press (2005)
(ADM gauge)
CFC metric
The second post-Newtonian deviation from isotropy
is the solution of
(Schäfer 1990)
(complicated) transverse, traceless projection
operator Newtonian potential
Modified equations for ?, ?i and ? (with respect
to CFC)
25
CFC metric equations (2)
We can solve the equations by introducing
some intermediate potentials
16 elliptic linear equations Linear solver LU
decomposition using standard LAPACK routines
Boundary conditions Multipole development in
compact-supported integrals
26
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
quasi-radial 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
27
CFCradial modes of spherical NS
quasi-radial modes of rotating NS
spherical NS
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)
28
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.
29
CFC core collapse dynamics (2)
Extreme case (torus-like structure)
Type II (multiple bounce)
30
CFC core collapse waveforms
Two distinct ways to extract waveforms
From the quadrupole formula
From the metric hij
Offset correction (dashed line)
Absolute differences between CFC and CFC
waveforms. No significant differences found.
31
Mariage des MaillagesHRSC 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 (high-order 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.
32
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.
2. Domain radius rd should stay fixed at roughly
   rpns after core bounce.
3. More than 3 domains needed in dynamical core
   collapse.
4. Compare with previous solvers in axisymmetry 33
   collocation points per domain sufficient.

rd held fixed (10 initial rse)
wrong result!
final rd too large
bounce time
only 3 radial domains
Gibbs-type oscillations
33
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)
3d low resolution 2d high
resolution 3d low resolution without artificial
perturbation
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.
Proof of principle code is ready for simulations
of dynamical triaxial instabilities.
34
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
35
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. 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 (DFM 02)
?
?W?6 (20 gain at bounce!)
A3B2G4 (A/rse0.32) Shibata Sekiguchi
(A/rse0.25)
A3B2G4
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.
Shibata Sekiguchi
36
CFC metric equations modification to allow for
black hole formation
Original CFC equations
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
To obtain one needs to first
compute the conformal factor, which is obtained
from the evolution equation
37
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.
38
Rotational core collapse to high-density NS CFC
vs CFC

• Model M7C5 (Shibata and Sekiguchi 2005)
• Differential rotation parameter A/R0.1
• Baryon rest mass M2.464
• Angular momentum J/M20.664
• Polytropic EOS (?4/3, k7x1014 (cgs))
• Excellent agreement with the full BSSN
  simulations of Shibata Sekiguchi (2005)

?max1.4x1015
GW amplitude larger at bounce with CFC
?min0.42
39
Core collapse simulations using the Einstein
equations for the Bondi metric
Reference Siebel, Font, Müller, and
Papadopoulos, PRD 67, 124018 (2003)
Ricci tensor
Hypersurface equations hierarchical set for
Evolution equation for
40
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 u0. 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
• Bondi news function (from the metric variables
  expansion at scri)
• Approximate gravitational waves (Winicour 1983,
  1984, 1987)
• Quadrupole news
• First moment of momentum formula

Null code test time of bounce
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).
41
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, 10-5-10-4c 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.
42
31 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 field measured by the observer
comoving with the fluid.
with the definitions
Ideal MHD condition electric four-corrent must
be finite.
43
31 General Relativistic (Ideal)
Magnetohydrodynamics equations (2)
Adding all up first-order,
flux-conservative, hyperbolic system of balance
laws constraint (divergence-free condition)

• Conservation of mass
• Conservation of energy and momentum
• Maxwells equations
• Induction equation
• Divergence-free constraint

44
Solution procedure of the GRMHD equations
Constrained transport scheme (Evans Hawley
1988, Tóth 2000). Field components defined at
cell interfaces. Zone-centered vector (needed for
primitive recovery and cell reconstruction
Riemann problem) obtained from staggered field
components

• Same HRSC schemes as for GRHD equations (HLL,
  Kurganov-Tadmor, Roe-type)
• Wave structure information obtained
• Primitive variable recovery more involved

Details Antón, Zanotti, Miralles, Martí, Ibáñez,
Font Pons, in preparation (2005)
Update of field components
These equations conserve the discretization of
45
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 t0.4 Open circles
nonvanishing lapse function (2), at time
t0.2 Open squares nonvanishing shift vector
(0.4), at time t0.16
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.
46
GRMHD equations code tests (2)
Magnetized spherical accretion onto a
Schwarzschild BH
density
Test difficulty keeping the stationarity of the
solution Used in the literature (Gammie et al
2003, De Villiers Hawley 2003) Initial data
Magnetic field of the type
on top of the hydrodynamic (Michel) accretion
solution. Radial magnetic field
component chosen to satisfy divergence-free
condition, and its strength is parametrized by
the ratio
internal energy
HLL solver 100 zones ?1 Solid lines analytic
solution Circles numerical solution
(t350M) Increasing the grid resolution shows
that code is second-order convergent irrespective
of the value of ?
radial velocity
radial magnetic field
47
GRMHD equations code tests (3)
Magnetized equatorial Kerr accretion (Takahashi
et al 1990, Gammie 1999)
density
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) 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.
azimuthal velocity
a0.5 FM-1.0 FL-2.815344 FE-0.908382 F??0.5 H
LL solver Solid lines analytic
solution Circles numerical solution (t200M)
radial magnetic field
azimuthal magnetic field
second order convergence
48
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.
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Summary of the talk

• Multidimensional simulations of relativistic
  core collapse feasible nowadays with current
  formulations of hydrodynamics and Einsteins
  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)