Numerical {3+1} General Relativistic Magnetohydrodynamics
L. A NT ÓN , O. Z ANOTTI , J.A. M IRALLES , J.M. M ART Í , J.M. I B Á ÑEZ , J.A. F ONT AND J.A. P ONS
1
1
2
1
ABSTRACT
where g is the determinant of the metric g ,

ρW
0

(ρh + b2 )W 2 vj − αb
bj

U=
(ρh + b2 )W 2 − p + 21 b2 − (αb0 )2
Bk
µν



S=


2
i
ds = gµν dx dx = −(α −βi β )dt +2βi dx dt+γij dx dx
where α (lapse function), β i (shift vector) and γij
(spatial metric) are functions of the coordinates t, xi .
Greek (Latin) indices run from 0 to 3 (1 to 3).
Relation between the velocity measured by the Eulerian observer v i and the four-velocity:
βi
ui
+ .
v =
αut
α
i
we define
βi
ṽ = v −
α
i

i
2
0k
The stationary solution of an ideal gas accreting onto a
Schwarzschild black hole, derived by Michel [6], is not
modified when adding a purely radial magnetic field. The
figure shows a simulation with B r = 10−2 /r2 at t =
213.2M . We use a grid with 200 × 8 zones, CFL parameter
0.4, and a Roe-type Riemann solver. The solid lines indicate
the analytic solution and the diamonds the numerical solution. Rs = 2GM/c2 is the Schwarzschild radius, M being
the mass of the black hole. Note that only one point out of
five are represented.
5 A torus around a black hole
5.1 Equilibrium torus without magnetic field
As initial model we use a rotating torus in equilibrium
around a Schwarzschild black hole of mass M . We run our
numerical code during more than four orbital periods of
the maximum density point (t=153 M ). A value of 1 for
the tracer corresponds to the fluid which was initially in
the torus. This simulation shows the capability of our code
in maintaining a stable initial configuration without significant changes.






Our numerical code is based on the following procedures:
• Conservative form:
To guarantee numerically the conservation of mass,
momentum and energy.
• Riemann solver:
Two alternative strategies to obtain the fluxes across
the numerical interfaces needed to integrate the evolution equations are used:
– A local characteristic approach [2] where local
Riemann problems are solved using a linearized
Riemann solver, namely the HLLE solver.
– An approach based on the equivalence principle
[7] in which a coordinate transformation to a locally Minkowskian spacetime is made, which allows to solve the Riemann problems using any
linearized Riemann solver for special relativistic
MHD.
• Induction equation and magnetic field constraint:
i
1
3 Numerical approach
The line element can be written as
i
0
T µν ∂µ gνj − Γδµν gδj
µ0
µν 0
α T ∂µ log α − T Γµν
Γµνλ being the Christoffel symbols.
• The {3 + 1} formalism
2



ρW ṽ
i
1 2
i
2
2
i

b
δ
−
b
bj 
(ρh
+
b
)W
v
ṽ
+
p
+
j
i
j
2


F =
1 2
2
2 i
0 i
i
(ρh + b )W ṽ − αb b − p + 2 b β /α 
B k ṽ i − B i ṽ k
2 GRMHD equations
ν

i

In many astrophysical scenarios, both magnetic and gravitational fields play an important role in determining the
evolution of the fluid. Examples of these include: a) neutron stars formation and binary neutron star coalescence,
most of which have intense magnetic fields of the order
of 1012 − 1013 G; b) magnetars, in which the intense magnetic fields and stresses may have important effects on
the internal structure of the star; c) relativistic jets, like
those observed in AGNs, microquasars, and gamma-ray
bursts, whose launching involves the hydromagnetic centrifugal acceleration of material from the accretion disk,
or the extraction of rotational energy from the ergosphere
of a Kerr black hole; d) accretion disks, whose magnetized plasma may undergo the magnetorotational instability, which plays an important role in the angular momentum transport throughout the disk. To study these interesting astrophysical objects it has become necessary to develop
numerical codes which allow us to describe and to evolve
the matter under such extreme conditions.
Our goal in this poster is to present the evolution equations
for the magnetic field and for the fluid within the {3+1} formalism, formulated in a suitable way to apply Godunovtype (or high-resolution shock-capturing; HRSC hereafter)
schemes based on (approximate) Riemann solvers.
µ
4.2 Magnetized Michel accretion
√
√
1 ∂ γU
1 ∂ −g Fi
√
=S
+√
i
−g ∂t
−g ∂x
1 Introduction
1
Departament d’Astronomia i Astrofı́sica, Universitat de València, Spain
2
Departament de Fı́sica Aplicada, Universitat d’Alacant, Spain
• Conservation equations
We describe the status of development of a numerical code
to solve the equations of ideal general relativistic magnetohydrodynamics (GRMHD). The numerical code, based
on high-resolution shock-capturing techniques, solves the
equations written in conservation form and computes the
numerical fluxes using a linearized Riemann solver. A special procedure is used to enforce the conservation of magnetic flux along the evolution. To show the capabilities
of the code, we present results of various tests, including
shock tubes in flat spacetime and magnetized fluid accretion onto a Schwarzschild black hole.
2
1
j
The induction equation is evolved using the procedure developed by [4], which has been adapted for
linearized Riemann solvers in the context of classical
MHD [8]. This method guarantees the divergence free
condition along the evolution.
5.2 Magnetized torus
4.3 Takahashi-Gammie accretion
Stationary solution of an ideal cold gas on the equatorial
plane accreting onto a Kerr black hole [9, 5, 3]. We use a grid
with 2000 radial cells and 5 angular cells subtending a small
angle of 10−4 π around the equatorial plane. The Riemann
solver used is HLLE and the figure shows a snapshot after
10000 time steps. Note that only one point out of five are
represented.
Now we add weak poloidal magnetic field loops on top
of the previous purely hydrodynamical equilibrium model.
Contrary to the non-magnetized case, this torus becomes
unstable due to the magneto-rotational instability. The figure shows a snapshot at the same time as the previous
model. By that time an important accretion flow has been
developed. The expansion and distortion of the initial torus
is apparent from the tracer panel.
• Time advancing:
We use a second/third order Runge-Kutta Scheme.
• Cell reconstruction:
In order to improve the spatial accuracy we use a
linear reconstruction method, namely the MINMOD
method, to obtain the values of the variables at the interfaces.
Lorentz factor
p
t
W = αu = 1/ (1 − v 2 )
with v = v · v = γij v v .
2
i j
• The Magnetic field
For ideal MHD the electromagnetic tensor is
F µν = −η µνλδ uλ bδ
where bα is the magnetic field measured by the comoving observer and η µνλδ is the volume element tensor.
Magnetic field measured by the Eulerian observer:
B i = α(bi u0 − b0 ui )
.
4 Numerical tests
4.1 Shock tube problem
Relativistic Brio & Wu problem [1] The test was performed
using a Cartesian grid of 1600 cells, CFL parameter 0.5 and
the HLLE Riemann solver in the local characteristic approach. Solid line corresponds to Minkowski spacetime at
t = 0.4. The triangles indicate the solution obtained with
α = 1, βx = 0.4 at t = 0.4. As we can observe, this solution
is shifted to the left a distance equal to βx × t in complete
accordance to the expected shift. The diamonds correspond
to the solution with α = 2.0, βx = 0 at t = 0.2. Again, this
solution is in complete agreement with the solution for the
Minkowski spacetime at t = 0.4.
References
[1] Balsara, 2001, ApJ, 132,83.
[2] Banyuls F., Font J.A., Ibáñez J.M., Martı́ J.M. & Miralles J.A.,
1997, ApJ, 476, 111.
• The stress-energy tensor
T µν
1
= (ρh + b2 )uµ uν + p + b2 g µν − bµ bν
2
where ρ is the rest-mass density, p is the pressure and
h is the specific enthalpy.
• Magnetic field constraint
√
1 ∂ γ Bi
∇·B= √
=0
i
γ ∂x
γ is the determinant of the spatial metric.
[3] De Villiers J.P. & Hawley J.F., 2003, ApJ, 589, 458.
[4] Evans C.R. & Hawley J.F., 1988, ApJ, 332, 659.
[5] Gammie C.F., 1999, ApJ, 522, L57.
[6] Michel F.C., 1972, Ap&SS, 15, 153.
[7] Pons J.A., Font J.A., Ibañez J.M., Martı́ J.M. & Miralles J.A.,
1998, A&A, 339, 638.
[8] Ryu D., Miniati F., Jones T.W. & Frank A., 1998, ApJ, 509, 244.
[9] Takahashi M., Nitta S., Tatematsu Y. & Tomimatsu A., 1990,
ApJ, 363, 206.