24P155-R
Beyond the Relativistic Magnetohydrodynamics: Numerical Scheme for
Relativistic Resistive Radiation Hydrodynamics
相対論的抵抗性輻射磁気流体コードの数値解法
Hiroyuki R. Takahashi, Ken Ohsuga
高橋 博之、大須賀健
Center for Computational Astrophysics (CfCA), National Astronomical Observatory of Japan (NAOJ)
2-21, Ohsawa, Mitaka, Tokyo 181-8588, Japan
国立天文台天文シミュレーションプロジェクト 〒181-8588 三鷹市大沢 2-21-1
We developed a numerical scheme for solving a set of Relativistic Resistive Radiation
Magnetohydrodynamics (R3MHD) equations, which ensures a conservation of total energy and momentumof the matter, magnetic field and radiation. The electric resistivity is assumed to be istropic in thecomoving frame. The radiation field is described by the 0th and 1st moment equations of the radiationtransfer equation. The hyperbolic terms, whose timescale is characterized by the light crossing time in therelativistic plasma, are explicitly integrated using approximate Rieman solvers. Source terms, describing amagnetic diffusion and an exchange of energy and momentum between the plasma and radiation, areimplicitly integrated using iteration method. This allows us to take a larger Cournat-Friedrichs-Lewyconditions when the magnetic Reynolds number is large or the plasma is optically thick. Our newly
developed scheme would be applicable for a number of relativistic astrophysical phenomena.

1. Introduction
Relativistic flows are appeared in many
astrophysical phenomena, such as the jets frommicroquasars and active galactic nuclei, pulsar
winds, soft gamma-ray repeaters, core-collapse
supernovae, and gamma-ray bursts. In many ofthese systems, the magnetic field has a crucialrole in their dynamics. For example, the magnetic
field connecting between the central star and theaccretion disk, or, the different points of theaccretion disks, are twisted and amplified due tothe differential rotation, launching the jets [1-3].
The jets are also powered by the rotational energyof the blackhole via the Poynting flux [4]. The
electric resistivity can play an important in these
dynamics since it changes the topology of themagnetic fields. The finite electric resistivity canbe the origin of the flares observed in the highenergy astrophysical phenomena [5-6].
The radiation field is also a key ingredient todetermine the dynamics of relativisticphenomena. The radiation pressure force canfacilitate the jet acceleration [7-9]. Recently theproduction of magnetically collimated,radiatively accelerated jet is modeled using nonrelativistic radiation magnetohydrodynamic
simulations [10]. Thus it is important toconsistently take into account these effects tounderstand the relativistic phenomena.
In the Relativistic Resistive Radiation
Magnetohydrodynamics (R3MHD), there are fourtimescales in the system: (i) the dynamical timesale, tdyn (ii) light crossing time tc, (iii) damping
timescale of electric fields t damp , and (iv)

absorption/emission or scattering timescales, taband tsc. The first two timescales are comparable in
the relativistic fluids. The damping time scale
tdamp, characterizing the magnetic diffusion due tothe ohmic dissipation, can be much shorter than
the former two timescales when the magnetic
Reynolds number is larger than unity. Also tab andtsc can be much shorter than tdyn and tc in the
optically thick medium. These facts indicate that
R3MHD equations become stiff for the highmagnetic Reynolds number and the large opticaldepth. Such stiff equations are difficult toexplicitly integrate with time. In this paper, wedeveloped an Explicit-Implicit scheme forsolving R3MHD equations to overcome theseproblems. This is the first challenge for theR3MHD simulations in the world.
2. Outline of numerical scheme
In the following, we take the light speed as unity.
The set of R3MHD equations is given by

where ρ, uν, TMHDµν, Tradµν, Fµν, Iν, σ, q0, are theproper mass density, bulk four velocity, energy
momentum tensors of magnetofluids and radiation,

Maxwell tensor of electromagnetic fields, fourvector of the electric current, conductivity, andproper electric charge density, respectively. Gν is theradiation four force. Equation (1) - (5) describes thetime evolution of the systems, while equation (6)gives the Ohm’s law. Another relation for theradiation field is needed to close the system, i.e., theclosure relation. We utilize the general form of theclosure relation as,

the absorption timescale in the clump is about 10
times shorter than the light crossing time. Thismeans that the numerical solution can be unstablewhen the source term is explicitly integrated. In ournumerical scheme, the implicit integration of source
term allows us to take a large time step without
numerical instability.

where Prij and Er are the radiation stress and theradiation energy density, while Dij is the so-called
Eddington tensor, which is generally a function ofradiation fields.
In our treatment, equation (1) - (5) is integratedusing an operator-splitting method. Then, theseequations are symbolically expressed bywhere U, F, S denote the conservative variable, flux
and source term, respectively. The source term
describes the magnetic diffusion and the gasradiation interaction so that equation (9) can be
stiff. We propose that equation (7) is explicitlyintegrated, while equation (9) is implicitlyintegrated using iteration method. The implicit
scheme allows us to take the numerical time step δtlarger than tdamp, tab and tsc.
3. Results and Conclusions.
The first numerical test is the so-called shadow
problem. The computation is performed in the 2dimensional Cartesian coordinate. The gas
distributes uniformly in space and the radiation
field equilibrates with the gas. The optical depth of
the system is about 0.01. There is a clump aroundthe origin that the plasma density is 1000 times
larger than that of the uniform matter. The
corresponding optical depth is about 10. The
radiation is injected at the left boundary (x=-5cm)
with a constant luminosity. Figure 1 shows thecolour contour of the radiation energy at the finalstate. In the upper panel, the Eddington
approximation is assumed in the closure relationthat the radiation field is isotropic in the momentum
space. In the lower panel, the M-1 closure [10] isadopted, which allows the anisotropy of theradiation fields. We can see that the radiationenergy density is uniform in space outside theclump with the Eddington approximation, while theshadow is successively reproduced behind theclump with the M-1 closure. We have to note thatFig.1. Numerical results of the shadow problems. Upperand lower panels correspond to results with theEddington approximation and the M-1 closure. Colour
shows the radiation energy density.

Acknowledgments
Numerical computations were carried out on
Cray XT4 at the Center for Computational
Astrophysics, CfCA, at the National Astronomical
Observatory of Japan, on Fujitsu FX-1 at the JAXASupercomputer System (JSS) at the Japan
Aerospace Exploration Agency (JAXA), and on
T2K at the University of Tokyo. This work is
supported in part by Ministry of Education, Culture,Sports, Science, and Technology (MEXT) for
Research Activity Start-up (HRT) 23840045 and forYoung Scientist (KO) 20740115. A part of thisresearch has been funded by MEXT HPCISTRATEGIC PROGRAM.
References[1] Lovelace R. V. E.: Nature 262 (1976) 649[2] Blandford R. D., & Payne D. G.: MNRAS 199(1982) 883[3] Uchida Y., & Shibata K.: PASJ 37 (1985) 515[4] Blandford R. D., & Znajek R. L.: MNRAS 179(1977) 443[5] Drenkhahn, G.: A&A, 387 (2002) 714[6] Gill, R. & Heyl, J. S.: MNRAS, 407 (2010) 1926
[7] Bisnovatyi-Kogan, G. S. & Blinnikov, S. I.: A&A59 (1977) 111[8] Sikora, M., Sol, H., Begelman, M. C., & Madejski,G. M.: MNRAS 280 (1996) 781[9] Takeuchi, S., Ohsuga, K., & Mineshige, S.: PASJ 62(2010) L43+
[10] Levermore, C. D.: J. Quant. Spec. Radiat. Transf. 31(1984) 149