Numerical Relativistic Hydrodynamics and
Magnetohydrodynamics, and Extragalactic Jets
José Mª Martí
Departamento de Astronomía y Astrofísica
Universidad de Valencia (Spain)
Astrophysical Fluid Dynamics, April 4-9, 2005
Outline of the talk
• Motivation. Extragalactic jets
•Discovery and observational status
•Standard model
•The jet/outflow family
• Extragalactic jets: open questions
• General relativistic hydrodynamics
•RHD eqs. as a hyperbolic system of conservation laws
•Numerical methods: AV, HRSC, …
• Simulations of extragalactic jets
• Large scale jets
• Compact jets
• Jet formation mechanisms
• Relativistic Magnetohydrodynamics
• Summary, conclusions, …
Motivation. Extragalactic jets
General relativity and relativistic hydrodynamics play a major role
in the description of many astrophysical scenarios. Extragalactic
jets (present in radio-loud AGNs) are a paradigmatic example.
1918: Discovery of the optical jet in M87
1940-50: Radio source Virgo A associated with M87 and its jet
1970-80: Models of continuous supply to interpret double lobed
extragalactic radio sources (obtained with aperture synthesis
techniques)
M87, HST
Scheuer 1974
Blandford & Rees 1974
1980-…: with the advent of VLA, routine detection of jets
Very Large Array
Socorro, NM
Cygnus A, Cambridge 5km array
Hardgrave & Ryle 1974
Cygnus A, VLA, Perley et al. 1984
Extragalactic jets: Observational
status I
Nowadays, jets are a common ingredient of radio-loud AGNs detected and imaged at very different
spatial scales with different arrays (kpc scales: VLA, Merlin; pc scales: VLBA, VLBI, VSOP)
VLBA
Extragalactic jets: Observational status
II
Kpc scales: Morphological dichotomy
based on jet power
Pc scales: Superluminal
motion, one-sidedness
Subpc scale: Collimation
Cyg A, VLA (6cm), Carilli et al. 1996
3C120, VLBA
Gómez et al. 2000
3C31, VLA (20cm), 1999
M87, VLA/VLBA
Junor et al. 1999
Extragalactic jets: Observational status
III
Radio polarization: information about
the topology of magnetic fields at both
pc and kpc scales
1055+018, Attridge et al. 1999
Hints on jet internal structure: shocks,
shear layers
3C31, Laing & Bridle 2002 (ticks show the orientation
of magnetic field
3C120: X-ray dips/component
ejection correlations
Variability: time scales and correlations
between variations in different
continuum/line components
Nature and location of the different
components at the central regions of
AGNs (e.g., narrow and broad line
regions) and on their interdependencies
(e.g., jet/disc connection)
Marscher et al. 2002
Extragalactic jets: Observational status
IV
Imaging of the central regions of AGNs: anatomy (narrow line region, obscuring torus) and
dynamics (rotation) of the central engine
NGC5728
NGC4261
Hubble Space Telescope
Multiwavelength imaging of the jets and spectra: emission models, spectral evolution of the high
energy electron distribution, reacceleration sites, …
Bertsch (NASA/GSFC)
3C273, radio (MERLIN), optical (HST) and X-ray (CHANDRA)
with optical contours; Marshall et al. 2001
Extragalactic jets: the standard model
The production of jets is connected with the process of
accretion on supermassive black holes at the core of AGNs
(see, e.g., Celotti & Blandford 2000)
• Hydromagnetic acceleration of disc wind in a BH
magnetosphere (Blandford-Payne mechanism)
• Extraction of rotational energy from Kerr BH by
magnetic processes (Blandford-Znajek mechanism,
magnetic Penrose process)
Emission: synchrotron (responsible of the emission from
radio to X-rays) and inverse Compton (g-ray emission)
from a relativistic (e+/e-, ep) jet (e.g., Ghisellini et al. 1998). Target
photons for the IC process:
•Self Compton: synchrotron photons
•External Compton: disc, BLR, dusty torus
Jets are relativistic, as indicated by:
•Superluminal motions at pc scales (due to the finite speed of propagation of light)
•One-sidedness of pc scale jets and brigthness asymmetries between jets and counterjets at
kpc scales (due to Doppler boosting of the emitted radiation)
Jets: Relativistic collimated ejections of thermal (e+/e-, ep) plasma + ultrarelativistic
electrons/positrons + magnetic fields + radiation, generated in the vicinity of SMBH
(GENERAL) RELATIVISTIC MHD + ELECTRON TRANSPORT + RADIATION TRANSFER
The jet/outflow
family
X-ray binaries:
• 20%
• vjet < 0.9c
• Ljet = 1038- 1040 erg/s
• Size = 10-3 – 10-1 pc
• Collimation: few degrees
• Central engine: stellar
BH/NS + disc
Active galactic nuclei:
• 10% AGN (radio-loud AGN)
• vjet = 0.995c
• Ljet = 1043- 1048 erg/s
• Size = 0.1-1 Mpc
• Collimation: few degrees
• Central engine: SMBH + disc
Young stellar objects:
• vjet = 10-3 c
• Ljet = 1035 erg/s
• Size = 10-3 – 10-2 pc
• Collimation: few degrees
• Central engine: YSO + inflow
And more… (pulsars, proto-planetary
nebulae, cataclismic variables,…)
Gamma-ray bursts:
• vjet/wind = 0.99995-0.999999995c
• LGRB = 1052 erg/s (T = 1s)
• Size = 1 pc (late afterglow evolution)
• Collimation: few tens of degrees
• Central engine: stellar BH + torus
miliparsec scale
Extragalactic jets: open
questions
Jet formation: hydromagnetic acceleration of disc wind ?
extraction of rotational energy from rotating BH
Influence of radiative acceleration
Influence of hydrodynamic acceleration
Origin of the poloidal magnetic field
Connection with jet
composition: ep, e+e-
Acceleration: present numerical simulations fail to generate highly relativistic, steady jets
(several arguments point to Lorentz factor  few-20; IDV: Lorentz factor 100)
Jet composition
Kiloparsec scale
parsec scale
Origin of the ultrarelativistic particle distribution
Nature of the radio components: relativistic shocks ?
instabilities ?
Structure and kinematics of jets; magnetic field topology; role of the magnetic fields in the
jet dynamics and emission
Stability on large scales
FRI/FRII morphological dichotomy: environment? Jet power? Composition? Formation
mechanism? Accretion regime? Magnetic field?
KH instabilities?
Role in galaxy and cluster evolution: heating
RHD equations for a perfect
fluid
The fluid is characterized by a four-velocity um and a stress-energy tensor Tmn. The space-time
in which the fluid evolves is characterized by a metric tensor whose components are gmn.
The equations of relativistic hydrodynamics are conservation laws:
• Conservation of mass:
r: proper rest-mass density
• Conservation of energy and momentum:
For a perfect fluid:
e: specific internal energy; p: pressure
• Equation of state:
Approaches:
Full GRHD: Components of the metric as a function of coordinates from Einstein eqs.
(
) (Numerical Relativity; consistent scenarios of jet/GRB production)
Test GRHD: Fluid evolving in a given space-time (simplified scenarios of jet formation)
SRHD: Flat space-time (propagation of jets at parsec and kiloparsec scales)
Relativistic hydrodynamics: SRHD equations
Relativistic hydrodynamics:
hyperbolicity
RHD equations form a non-linear, hyperbolic system of conservation laws for causal
EoS (e.g., Anile 1989)
For hyperbolic systems,
• The Jacobians of the vectors of fluxes,
complete set of eigenvectors.
, have real eigenvalues and a
• Information about the solution propagates at finite velocities (characteristic speeds)
given by the eigenvalues of the Jacobians.
• Hence, if the solution is known (in some spatial domain) at some given time, this fact
can be used to advance the solution to some later time (initial value problem).
• However, in general, it is not possible to derive the exact solution for this problem.
Instead, one has to rely on numerical methods which provide an approximate to the
solution.
• Moreover, these numerical methods must be able to handle discontinuous solutions,
which are inherent to non-linear hyperbolic systems.
Relativistic hydrodynamics: characteristic structure I
Donat et al. 1998; Martí et al. 1991, Eulderink 1993, Font et al. 1994
Relativistic hydrodynamics: characteristic structure II
Numerical relativistic hydrodynamics: AV
methods I
May & White’s code (May & White 1966, 1967)
Lagrangian (1D) finite difference scheme + artificial viscosity (AV) for spherically symmetric, relativistic, stellar collapse
Space-time metric:
m: total (barionic) rest-mass up to radius R
Mass, energy and
momentum conservation:
Einstein equations:
EoS:
Richtmyer & von Neumann’s (1950) AV
AV produces dissipation
at shocks that reduces
post-shock oscillations
Numerical relativistic hydrodynamics: AV
methods II
Wilson’s approach (Wilson 1972, 1979)
Eulerian 2D finite difference scheme + artificial viscosity (AV) for test GRHD (3+1formalism)
Mass conservation:
Momentum conservation:
Energy conservation:
EoS:
Extension to Full GRHD: Smarr & Wilson code (Wilson 1979): Wilson’s formulation built on a
vacuum numerical relativity code for the head-on collision of two black holes (Smarr 1975)
Smarr & Wilson’s approach allowed to simulate complex relativistic scenarios for the first time:
• Axisymmetric stellar core collapse (Wilson 1979, Dykema 1980, Nakamura et al. 1980, Bardeen & Piran
1983, Evans 1984, …)
• Issues on Numerical Cosmology: inflation, primordial nucleosynthesis, microwave
anisotropy, evolution of primordial gravity waves,…(Centrella & Wilson 1983, 1984, …)
• Accretion onto compact objects (Hawley et al. 1984a,b, …)
• Heavy ion collisions (Wilson & Mathews 1989, …)
Numerical relativistic hydrodynamics: AV methods
III
Wilson’s approach. Numerical method (Wilson 1972, 1979; see the recent book by Wilson & Mathews 2003)
Eulerian 2D finite difference scheme + artificial viscosity (AV) for test GRHD (3+1formalism)
Model equation in 1D:
(transport equation with source terms)
Discretization:
(Donnor cell or Lelevier’s method;
upwind, first order)
Newer developments: Consistent AV
Non-consistent AV: artificial viscosity
added to the pressure in some terms of
the hydro eqs.
Large innacuracies in mildly relativistic flows
artificial viscosity as a bulk scalar viscosity
(Norman & Winkler 1986):
T mn  rhu m un  pg mn    g mn  u m un )
Very accurate; increase of coupling
High-Resolution Shock-Capturing methods
see LeVeque’s 1992 book
HRSC methods deal with hyperbolic systems
of conservation laws. Model equation in 1D:
HRSC methods: finite difference (volume) schemes in conservation form. Integrating the PDE
over a finite space-time domain
:
Lax-Wendroff theorem (LW 1960): conservation form ensures the convergence of the solution
under grid refinement to one of the weak solutions of the original system of equations
Numerical fluxes: - exact or approximate Riemann solvers (Godunov-type methods)
- standard finite-difference methods + local conservative dissipation terms
(e.g., symmetric schemes)
High-order of accuracy:
• Total variation stable (TVD, TVB) algorithms
• Standard approach: Conservative monotonic
polynomials as interpolant functions within
zones (slope limiter methods; also flux limiter
methods)
Stability (no spurious oscillations)
and convergence
• MINMOD (van Leer 1977; second order)
• PPM (Colella & Woodward 1984; third order)
• ENO (Harten et al. 1987)
HRSC methods in Relativistic Hydrodynamics
See Martí & Müller, Numerical Hydrodynamics in Special Relativity,Living Reviews in Relativity,
http://www.livingreviews.org/Articles/lrr-2003-7
Based on Riemann solvers (upwind methods):
• Linearized solvers: based on local linearizations
of the Jacobian matrices of the vector of fluxes
– Roe-type Riemann solvers (Roe-Eulderink:
Eulderink 1993; LCA: Martí et al. 1991)
– Falle-Komissarov (Falle & Komissarov 1996):
based on a primitive-variable formulation of
the eqs.
– Marquina Flux Formula (Donat & Marquina 96)
• Solvers relying on the exact solution of the
Riemann problem (Martí & Müller 1994; Pons, Martí & Müller
2000)
– rPPM (Martí & Müller 1996)
– Random choice method (Wen et al. 1997)
– Two-shock approximation (Balsara 1994; Dai
& Woodward 1997)
HRSC METHODS DESCRIBE ACCURATELY
HIGHLY RELATIVISTIC FLOWS WITH STRONG
SHOKS AND THIN STRUCTURES
Symmetric TVD, ENO schemes with nonlinear numerical dissipation:
• LW scheme with conservative TVD dissipation terms (Koide at al. 1996)
• Del Zanna & Bucciantini 2002: Third-order ENO reconstruction algorithm + spectraldecomposition-avoiding RS (LF, HLL)
• NOCD (Anninos & Fragile 2002)
Exact Riemann solver in RHD I
(Martí & Müller 1994; Pons, Martí & Müller 2000)
Riemann Problem: IVP with initial discontinuous data L, R
W: shock / rarefaction (self-similar expansion); C: contact discontinuity
1. The compressive character of shock waves allows as to
discriminate between shocks (S) and rarefaction waves (R):
pressure ahead and behind the wave
2. The functions Wg, Wf allow one to determine the functions
v x R* ( p) and v x L* ( p) , respectively.
RS(p) / SS(p): family of all states which can be connected through a rarefaction / shock with a given state S
ahead the wave.
3. The pressure p* and the velocity vx* in the intermediate states are then given by the condition
across the contact discontinuity
Exact Riemann solver in RHD
II
Solution of the Riemann problem in the pressure – normal flow velocity diagram
(Martí & Müller 1994,
Pons, Martí & Müller 2000)
v x L* ( p)
v x R* ( p)
Intrinsic relativistic effects:
Riemann solution depends
on tangential velocities
solution for pure normal flow
The wave-pattern (RR, SR, SS) of the solution can be predicted in terms of the relativistic invariant relative
velocity between the initial left and right states (Rezzolla & Zanotti 2001,Rezzolla, Zanotti & Pons 2002)
S
Relative velocity
between L, R initial
states
changing initial
tangential speeds…
S
R
S
Algorithms for Numerical RHD I: upwind HRSC methods
Algorithms for Numerical RHD II: central HRSC
methods
Harten-Lax-van Leer Flux
Algorithms for Numerical RHD III: other approaches
Special relativistic Riemann solvers in GR
Pons et al. 1998
According to the Equivalence Principle, physical laws in a local inertial frame* of a curved
spacetime have the same form as in Special Relativity
* free falling frames
The solution of the Riemann problem in GR coincides (locally) with the one in SR if
described in a LIF
1. Perform at each numerical interface,  x 1 , a coordinate transformation to locally Minkowskian
coordinates, x̂, according to
xˆ M  ( x  x0 )
ˆ
ˆ
where x0 are the coordinates of the center of the interface and M is given by
 M  eˆ
ˆ
eˆ  eˆ ˆˆ
where e̂ is the orthonormal basis attached to x̂ , with e1̂ orthogonal to  1 .
x
2. Set up the Riemann problem at the two sides of  1 by transforming the velocity
x
components to the new basis, e̂ .
3. Solve the Riemann problem as in special relativity and compute numerical fluxes
4. Transform numerical fluxes to the coordinate basis,  .
This procedure has been used with success in numerical GRMHD and force-free degenerate electrodynamics
in the context of jet formation scenarios (Komissarov 2001, 2004, 2005)
Simulations of relativistic jets: Kiloparsec scale
jets I
Hydrodynamical non-relativistic simulations (Rayburn 1977; Norman et al. 1982) verified the basic
jet model for classical radio sources (Blandford & Rees 1974; Scheuer 1974).
Two parameters control the morphology and dynamics of jets: the beam to external density ratio and the
internal beam Mach number
Morphology and dynamics
governed by interaction with
the external (intergalactic)
medium. The simulations have
allowed to identify the
structural components of radio
jets
First relativistic simulations: van Putten 1993, Martí et al. 1994, 1995, 1997; Duncan & Hughes 1994
Relativistic, hot jet models
Relativistic, cold jet models
Density + velocity field vectors
“featureless” jet + thin cocoons without backflow
+ stable terminal shock: naked quasar jets (e.g.,
3C273)
“knotty” jet + extended cocoon + dynamical
working surface: FRII radio galaxies and lobe
dominated quasars (e.g., Cyg A)
Simulations of relativistic jets: Kiloparsec scale
jets II
3D simulations (Nishikawa et al. 1997, 1998; Aloy et al.
1999; Hughes et al. 2002, …):
• Simulations too short (mean jet advance speed
too high; poorly developed cocoons)
• Two-component jet structure: fast (LoF  7)
inner jet + slower (LoF  1.7) shear layer with
high specific internal energy
8.6 Mcells
3.6 Mcells
Aloy et al. 1999
Long term evolution and jet composition (Scheck et al. 2002):
• Evolution followed up to 6 106 y (10% of a realistic
lifetime).
• Realistic EoS (mixture of e-, e+, p)
• Long term evolution consistent with that inferred for
powerful radio sources
• Relativistic speeds up to kpc scales
• Neither important morphological nor evolutionary
differences related with the plasma composition
Simulations of relativistic jets: RMHD sims of kpc
jets
(Nishikawa et al. 1997, 1998; Komissarov 1999; Leismann et al. 2005)
Relativistic jet propagation along aligned and oblique magnetic fields (Nishikawa et al. 1997, 1998)
Relativistic jets carrying toroidal magnetic fields (Komissarov 1999):
• Beams are pinched
• Large nose cones (already discovered in classical MHD
simulations) develop in the case of jets with Poynting flux
• Low Poynting flux jets may develop magnetically confining
cocoons (large scale jet confinement by dynamically
important magnetic fields)
Models with poloidal magnetic fields (Leismann et al. 2005):
•The magnetic tension along the jet affects the
structure and dynamics of the flow.
•Comparison with models with toroidal magnetic fields:
-The magnetic field is almost evacuated form the
cocoon. Cocoons are smoother.
- Oblique shocks in the beam are weaker.
Simulations of relativistic jets: pc-scale jets and superluminal radio
sources
Shok-in-jet model: steady relativistic jet with finite opening angle + small perturbation (Gómez et al. 1996,
1997; Komissarov & Falle 1996, 1997)
Relativistic perturbation
Pressure-matched jet
steady jet
Overpressured jet
standing shocks
Radio emission (synchrotron; overpressured jet)
standing shocks
•Convolved maps (typical VLBI resolution; contours): core-jet structure
with superluminal (8.6c) component
• Unconvolved maps (grey scale):
- Steady components associated to recollimation shocks
- dragging of components accompanied by an increase in flux
- correct identification of components (left panel) based on the
analysis of hydrodynamical quantities in the observer’s frame
3D hydro+emission sims of relativistic precessing jets
(including light travel time delays): Aloy et al. 2003
Relativistic hydrodynamics and emission models
In order to compare with observations, simulations of parsec scale jets must account for
relativistic effects (light aberration, Doppler shift, light travel time delays) in the emission
Basic hydro/emission coupling (only synchrotron emission considered so far! Gómez et al. 1995, 1997;
Mioduszewski et al. 1997; Komissarov and Falle 1997):
• Dynamics governed by the thermal (hydrodynamic) population
• Particle and energy densities of the radiating (non-thermal) and hydrodynamic populations proportional
(valid for adiabatic processes
• (Dynamically negligible) ad-hoc magnetic field with the energy density proportional to fluid energy density
• Integration of the radiative transfer equations in the observer’s frame for the Stokes parameters along the
line of sight
• Time delays: emission ( en ) and absortion coefficients ( n ) computed at retarded times
• Doppler boosting (aberration + Doppler shift):
e vobob   2en ,  vobob   1 v , 
 v ob / v   ( cos )
Further improvements:
• Compute relativistic electron transport during the jet evolution to acount for adiabatic and radiative losses
and particle accelerations of the non-thermal population (e.g., Jones et al. 1999, non-relativistic MHD sims.)
• Include inverse Compton (scattering process!)
• Include emission back reaction on the flow (important at high frequencies)
Simulations of superluminal sources:
interpreting the observations with the hydrodynamical shock-in-jet
model
Isolated (3C279, Wehrle et al. 2001) and regularly spaced stationary components (0836+710, Krichbaum
et al. 1990; 0735+178, Gabuzda et al. 1994; M87, Junor & Biretta 1995; 3C371, Gómez & Marscher 2000)
Variations in the apparent motion and light curves of components (3C345, 0836+71, 3C454.3, 3C273,
Zensus et al. 1995; 4C39.25, Alberdi et al. 1993; 3C263, Hough et al. 1996)
Coexistence of sub and superluminal components (4C39.25, Alberdi et al. 1993; 1606+106, Piner &
Kingham 1998) and differences between pattern and bulk Lorentz factors (Mrk 421, Piner et al. 1999)
Dragging of components (0735+178, Gabuzda et al. 1994; 3C120, Gómez et al. 1998; 3C279, Wehrle et
al. 1997)
Trailing components (3C120, Gómez et al. 1998, 2001; Cen A, Tingay et al. 2001)
Pop-up components (PKS0420-014, Zhou et al. 2000)
1606+106
Piner & Kingham 1998
3C371
Gómez & Marscher 2000
Gabuzda et al.1994
0735+178
3C263
Hough et al. 1996
3C120
Gómez et al. 1998
3C279
Wehrle et al. 1997
Kelvin-Helmholtz instabilities and extragalactic
jets
KH stability analysis is currently used to probe the physical conditions in extragalactic jets
Linear KH stability theory:
• Production of radio components
• Interpretation of structures (bends, knots) as signatures of
pinch/helical modes
Non-linear regime:
• Overall stability and jet disruption
• Shear layer formation and generation of transversal structure
• FRI/FRII morphological jet dichotomy
Interpretation of parsec scale jets
Wavelike helical structures with differentially moving and stationary
features can be produced by precession and wave-wave interactions
(Hardee 2000, 2001)
[used to constrain the physical conditions in the inner jet of 3C120
(Hardee 2003, Hardee et al. 2005)]
The 3C273 case
1. Emission across the jet resolved
double helix inside the jet
2. Five sinusoidal modes are required to fit the double helix
3. The sinusoidal modes are then identified with instability modes
(elliptical/helical body/surface modes) at their respective resonant
wavelengths from which physical jet conditions are derived:
Lorentz factor: 2.1  0.4; Mach number: 3.5 1.4
Density ratio: 0.023  0.012; Jet sound speed: 0.53  0.16
Kelvin-Helmholtz instabilities: non-linear regime
Goals of the study:
• Relativistic effects on the stability of jets
• Transition from linear to nonlinear stability
Numerical simulations:
• Planar (2D) symmetric (pinch) / antysymmetric
(helical) modes
• Resolution: 400 zones/Rj (transversal) x 16 zones/Rj
(longitudinal)
Perucho et al. 2004a,b
Perucho et al. 2005
• Dynamics of jet disruption
• Long term evolution (jet disruption, shear layer
formation, …)
• Initial conditions: steady jet + small amplitude
perturbation (first body mode)
• Temporal approach
Lorentz factor 5
model
Initial model
Linear phase
Non-linear evolution
Evolutionary phases:
Linear
Saturation
Pressure maximum
Mixing
Quasisteady
Simulations of jet formation (accretion/outflow, acceleration,
collimation)
Plasma acceleration in BP mechanism
Blandford-Payne mechanism: hydromagnetic acceleration of
disk wind in a BH magnetosphere (barion loaded jet; GRMHD)
Accretion/ejection from Keplerian (co-rotating/counter-rotating) cold
disks around Schwarzschild (Koide et al. 1997), rapidly rotating Kerr
BH (Koide et al. 2000)
Forces acting on
plasma particles
along B lines
Kerr BH
Koide et al. 2000
B line anchored
to the disk
Schwarzschild BH
Koide et al. 1997
particles tied
to B lines
(MHD approx.)
Magnetic Penrose process
Magnetic line twist extract
energy from BH ergosphere
Extraction of rotational energy form Kerr BH by magnetic processes:
• Blandford-Znajek mechanism: BH as a magnetized rotating conductor whose
rotational energy can be efficiently extracted by means of magnetic torque (Poynting
flux jet; force-free electrodynamics). Confirmed recently by Komissarov (2001 -FFDE-, 2004
-GRMHD, rarefied plasma-): formation of a UR particle, Poynting dominated wind
• magnetic Penrose process: magnetic field lines accross the ergosphere
twisted by frame dragging
line twist propagates outwards as torsional Alfven
wave train carrying e.m. energy
total energy of the plasma near the hole
decrease to negative values
swallowing of this plasma by the hole reduces the
BH rotational energy (Poynting flux jet; GRMHD). Koide et al. 2002, Koide 2003;
controversial: Komissarov 2005 (magnetic Penrose process does not operate…)
Koide et al. 2002
Relativistic Magnetohydrodynamics
RMHD as hyperbolic system of conservation laws
In one spatial dimension, the system of RMHD can be written as a hyperbolic system of
conservation laws for the unknowns
(conserved variables) subject to the
constraint
along the evolution.
Anile & Pennisi 1987, Anile 1989 (see also van Putten 1991) have studied the characteristic
structure of the equations (eigenvalues, right/left eigenvectors) in the space of covariant
variables
There are seven physical waves:
• Two Alfven waves, a , 
• Two fast magnetosonic waves,  f , 
• Two slow magnetosonic waves, s,
• One entropy wave, e
Wavefront diagrams in
the fluid rest frame
(Jeffrey & Taniuti 1964)
b n
b t  0
 f ,  a ,  s,  e  s,  a ,   f ,

The wave propagation
velocity depend on the
relative orientation of
the magnetic field, 
As in classical MHD there are two
kinds of degeneracies:
• Degeneracy I: b n  0

• Degeneracy II: b t  0
b
n
(Deg II)
b t
0
(Deg I)
orientation of the
magnetic field
• Physically: two or more wavespeeds become equal
(compound waves)
• Numerically: the spectral decomposition (needed in
upwind HRSC methods) blows up
Algorithms for Numerical RMHD
(HRSC+AV)
Numerical RMHD: 1D Tests
Fast, slow shocks & rarefactions; Alfvén waves; shock tubes*; magnetic field divergence-free condition
satisfied by construction
1.0
0.0
Compound wave
1.0
r
0.4
0.0
vx
-0.1
-0.8
1.0
1.5
p
0.0
1.0
vy
Deg I case
(Bx=0)
r
vx
30.0
0.0
p
e-12
0.0
0.0
20.0
2.0
vy
1.0
0.0
-1.5
1.0
0.2
0.0
0.0
* still lacking an analytical solution!!
1.0
By
0.0 Lorentz factor
Antón et al. 2005 (also van Putten 1993, Balsara 2001, Del
Zanna et al. 2002, De Villiers & Hawley 2003, …)
0.0
By
0.0
Lorentz factor
Antón et al. 2005 (also Komissarov 1999, Del Zanna et al.
2002, De Villiers & Hawley 2003, …)
Numerical RMHD: 2D Tests
Cylindrical explosions:
Ambient (r >1.0): p = 3.e-5, r = 1.e-4
(Smooth) transition layer (0.8 < r < 1.0)
Cylinder (r < 0.8): p = 1.0, r = 1.e-2
-3.1 log r + B lines
-1.5
-4.2
-4.6
Homogeneous magnetic field, B = (Bx, 0, 0)
Bx = 0.01, 0.1, 1.0 ( = 1.6, 1.6e2, 1.6e4)
4.7 Lorentz factor
log p
1.0
Antón et al. 2005 (proposed by Komissarov 1999; see also Del Zanna et al. 2002,  = 8.e2)
Rotors:
Static ambient (r > 0.1): p = 1.0, r = 1.0, B = (Bx, 0, 0), Bx = 1.0 ( = 0.5)
Rotating disk (r < 0.1): p = 1.0, r = 10.0, w= 9.95 (Lorentz factor approx. 10)
Del Zanna et al. 2002
t = 0.4
0.35 (w) – 8.19 (b)
5.3e-3 (w) – 3.9 (b)
3.8e-4 (w) – 2.4 (b)
1.0 (w) – 1.79 (b)
Summary, conclusions,…
• Extraordinary advance, in the last decade, in numerical methods for (ultra)relativistic
hydrodynamics, specially with RS/Sym HRSC methods
• Easy extension to (test) GRHD via local linearizations of the geometrical terms or local coordinate
transformations (for RS HRSC methods; Pons et al. 1998)
• Important advances in numerical RMHD (RS/Sym HRSC methods, also AV methods), however
present numerical codes are less robust than in the purely hydro case
• Big impact in extragalactic jet research, specially parsec scale jets, superluminal sources and jet
formation mechanisms
• Important advances in the understanding of the morphology and dynamics of large scale relativistic jets
• First simulations of superluminal sources (success of the relativistic shock-in-jet model). First steps in the
combination of hydro + relativistic electron transport + radiation transfer codes
• First simulations of relativistic jet formation
• Numerical study of the non-linear regime of KH instabilities
• Still lacking…
• RMHD sims of pc and kpc jets to elucidate the configuration and role of magnetic fields at these scales
• Jet formation mechanisms need further numerical study (no steady relativistic outflow yet found)
• Consistent simulation of all the jet components (thermal matter, high energy particles,
radiation and magnetic fields) and their mutual interaction.