Three-dimensional general relativistic radiation magnetohydrodynamical simulation of super-Eddington accretion, using a new code HARMRAD with M1 closure

Black hole (BH) accretion flows and jets are dynamic hot relativistic magnetized plasma flows whose radiative opacity can significantly affect flow structure and behavior. We describe a numerical scheme, tests, and an astrophysically relevant application using the M1 radiation closure within a new three-dimensional (3D) general relativistic (GR) radiation (R) magnetohydrodynamics (MHD) massively parallel code called HARMRAD. Our 3D GRRMHD simulation of super-Eddington accretion (about $20$ times Eddington) onto a rapidly rotating BH (dimensionless spin $j=0.9375$) shows sustained non-axisymmemtric disk turbulence, a persistent electromagnetic jet driven by the Blandford-Znajek effect, and a total radiative output consistently near the Eddington rate. The total accretion efficiency is of order $20\%$, the large-scale electromagnetic jet efficiency is of order $10\%$, and the total radiative efficiency that reaches large distances remains low at only order $1\%$. However, the radiation jet and the electromagnetic jet both emerge from a geometrically beamed polar region, with super-Eddington isotropic equivalent luminosities. Such simulations with HARMRAD can enlighten the role of BH spin vs.\ disks in launching jets, help determine the origin of spectral and temporal states in x-ray binaries, help understand how tidal disruption events (TDEs) work, provide an accurate horizon-scale flow structure for M87 and other active galactic nuclei (AGN), and isolate whether AGN feedback is driven by radiation or by an electromagnetic, thermal, or kinetic wind/jet. For example, the low radiative efficiency and weak BH spin-down rate from our simulation suggest that BH growth over cosmological times to billions of solar masses by redshifts of $z\sim 6-8$ is achievable even with rapidly rotating BHs and ten solar mass BH seeds.

[1]  Douglas M. Eardley,et al.  Black Holes in Binary Systems: Instability of Disk Accretion , 1974 .

[2]  Kip S. Thorne,et al.  Disk-Accretion onto a Black Hole. Time-Averaged Structure of Accretion Disk , 1974 .

[3]  K. Thorne Disk-Accretion onto a Black Hole. II. Evolution of the Hole , 1974 .

[4]  R. Blandford,et al.  Electromagnetic extraction of energy from Kerr black holes , 1977 .

[5]  S. Ichimaru Bimodal behavior of accretion disks: Theory and application to Cygnus X-1 transitions , 1977 .

[6]  T. Piran The role of viscosity and cooling mechanisms in the stability of accretion disks. , 1978 .

[7]  M. Begelman Can a spherically accreting black hole radiate very near the Eddington limit , 1979 .

[8]  H. Ford,et al.  The bizarre spectrum of SS 433. , 1979 .

[9]  Saul A. Teukolsky,et al.  Black Holes, White Dwarfs, and Neutron Stars , 1983 .

[10]  J. Papaloizou,et al.  The dynamical stability of differentially rotating discs with constant specific angular momentum , 1984 .

[11]  C. D. Levermore,et al.  Relating Eddington factors to flux limiters , 1984 .

[12]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[13]  John F. Hawley,et al.  A Numerical Study of Nonspherical Black Hole Accretion , 1984 .

[14]  Bruce Margon,et al.  Observations of SS 433 , 1984 .

[15]  O. Blaes Stabilization of non-axisymmetric instabilities in a rotating flow by accretion on to a central black hole , 1987 .

[16]  J. Lasota,et al.  Slim Accretion Disks , 1988 .

[17]  Spherical accretion onto black holes : a complete analysis of stationary solutions , 1991 .

[18]  Ramesh Narayan,et al.  Advection-dominated Accretion: A Self-similar Solution , 1994 .

[19]  Thermal equilibria of accretion disks , 1994, astro-ph/9409018.

[20]  R. Narayan,et al.  Advection dominated accretion: Underfed black holes and neutron stars , 1994, astro-ph/9411059.

[21]  Advection-Dominated Accretion Flows Around Kerr Black Holes , 1996, astro-ph/9607021.

[22]  R. Narayan,et al.  Advection-Dominated Accretion and the Spectral States of Black Hole X-Ray Binaries: Application to Nova Muscae 1991 , 1997 .

[23]  C. Gammie,et al.  Advection-dominated Accretion Flows in the Kerr Metric. II. Steady State Global Solutions , 1998, astro-ph/9802321.

[24]  Wei Cui,et al.  Spectral Transitions in Cygnus X-1 and Other Black Hole X-Ray Binaries , 1997, astro-ph/9711167.

[25]  J. Hawley,et al.  Instability, turbulence, and enhanced transport in accretion disks , 1997 .

[26]  Efficiency of Magnetized Thin Accretion Disks in the Kerr Metric , 1999, astro-ph/9906223.

[27]  Roger D. Blandford,et al.  On the fate of gas accreting at a low rate on to a black hole , 1998, astro-ph/9809083.

[28]  Global Simulations of Differentially Rotating Magnetized Disks: Formation of Low-beta Filaments and Structured Coronae. , 1999, The Astrophysical journal.

[29]  E. Quataert,et al.  Convection-dominated Accretion Flows , 1999, astro-ph/9912440.

[30]  B. Fryxell,et al.  FLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes , 2000 .

[31]  R. Narayan,et al.  Self-similar Accretion Flows with Convection , 1999, astro-ph/9912449.

[32]  James M. Stone,et al.  Magnetohydrodynamical non‐radiative accretion flows in two dimensions , 2001 .

[33]  J. Stone,et al.  A Magnetohydrodynamic Nonradiative Accretion Flow in Three Dimensions , 2001, astro-ph/0103522.

[34]  J. Krolik,et al.  Global MHD Simulation of the Inner Accretion Disk in a Pseudo-Newtonian Potential , 2000, astro-ph/0006456.

[35]  M. Mouchet,et al.  Are quasars accreting at super-Eddington rates? , 2002, astro-ph/0203439.

[36]  J. Hawley,et al.  The Dynamical Structure of Nonradiative Black Hole Accretion Flows , 2002, astro-ph/0203309.

[37]  M. Mori,et al.  Does the Slim-Disk Model Correctly Consider Photon-trapping Effects? , 2002, astro-ph/0203425.

[38]  Andrew King,et al.  Accretion Power in Astrophysics: Contents , 2002 .

[39]  Three-dimensional Magnetohydrodynamic Simulations of Spherical Accretion , 2001, astro-ph/0105365.

[40]  Axel Klar,et al.  Half-moment closure for radiative transfer equations , 2002 .

[41]  J. Krolik,et al.  Magnetically Driven Accretion Flows in the Kerr Metric. I. Models and Overall Structure , 2003, astro-ph/0307260.

[42]  S. Mineshige,et al.  Spectral Energy Distribution in Supercritical Disk Accretion Flows through Photon-trapping Effects , 2003, astro-ph/0306546.

[43]  Michael L. Norman,et al.  Beyond Flux-Limited Diffusion: Parallel Algorithms for Multidimensional Radiation Hydrodynamics , 2003 .

[44]  Ramesh Narayan,et al.  Nonthermal Electrons in Radiatively Inefficient Accretion Flow Models of Sagittarius A* , 2003, astro-ph/0304125.

[45]  Axel Klar,et al.  A half space moment approximation to the radiative heat transfer equations , 2003 .

[46]  S. Shapiro,et al.  Black Hole Spin Evolution , 2003, astro-ph/0310886.

[47]  Local Three-dimensional Simulations of Magnetorotational Instability in Radiation-dominated Accretion Disks , 2003, astro-ph/0304511.

[48]  L. Ho,et al.  Iron Emission in the z = 6.4 Quasar SDSS J114816.64+525150.3 , 2003, astro-ph/0308005.

[49]  Astrophysics,et al.  The Fate of Nonradiative Magnetized Accretion Flows: Magnetically Frustrated Convection , 2003, astro-ph/0304227.

[50]  E. Müller,et al.  Numerical Hydrodynamics in Special Relativity , 1999, Living reviews in relativity.

[51]  Charles F. Gammie,et al.  HARM: A NUMERICAL SCHEME FOR GENERAL RELATIVISTIC MAGNETOHYDRODYNAMICS , 2003 .

[52]  R. Narayan,et al.  Three-dimensional MHD Simulations of Radiatively Inefficient Accretion Flows , 2003, astro-ph/0301402.

[53]  Super-Eddington black hole accretion: Polish doughnuts and slim disks , 2004, astro-ph/0411185.

[54]  The Stability of Magnetized Rotating Plasmas with Superthermal Fields , 2004, astro-ph/0406071.

[55]  A. Klar,et al.  Multigroup half space moment approximations to the radiative heat transfer equations , 2004 .

[56]  T. D. Matteo,et al.  Modelling feedback from stars and black holes in galaxy mergers , 2004, astro-ph/0411108.

[57]  T. Belloni,et al.  GRS 1915+105 and the Disc-Jet Coupling in Accreting Black Hole Systems , 2004 .

[58]  C. Gammie,et al.  A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole , 2004, astro-ph/0404512.

[59]  E. Quataert,et al.  On the Maximum Luminosity of Galaxies and Their Central Black Holes: Feedback from Momentum-driven Winds , 2004, astro-ph/0406070.

[60]  Rodolphe Turpault A consistent multigroup model for radiative transfer and its underlying mean opacities , 2005 .

[61]  Imaging of sdss z > 6 quasar fields : Gravitational lensing, companion galaxies, and the host dark matter halos , 2005, astro-ph/0503202.

[62]  The Distribution and cosmic evolution of massive black hole spins , 2004, astro-ph/0410342.

[63]  Lorenzo Pareschi,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2010, 1009.2757.

[65]  M. Mori,et al.  Supercritical Accretion Flows around Black Holes: Two-dimensional, Radiation Pressure-dominated Disks with Photon Trapping , 2005, astro-ph/0504168.

[66]  Alan A. Wray,et al.  A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources , 2005 .

[67]  Santa Barbara,et al.  Cosmos++: Relativistic Magnetohydrodynamics on Unstructured Grids with Local Adaptive Refinement , 2005, astro-ph/0509254.

[68]  T. D. Matteo,et al.  Energy input from quasars regulates the growth and activity of black holes and their host galaxies , 2005, Nature.

[69]  K. Ohsuga Two-dimensional Radiation-Hydrodynamic Model for Limit-Cycle Oscillations of Luminous Accretion Disks , 2005, astro-ph/0512178.

[70]  Robert H. Becker,et al.  Constraining the Evolution of the Ionizing Background and the Epoch of Reionization with z ∼ 6 Quasars. II. A Sample of 19 Quasars , 2005, astro-ph/0512082.

[71]  W. Lewin,et al.  Compact stellar X-ray sources , 2006 .

[72]  C. Gammie,et al.  Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics , 2005, astro-ph/0512420.

[73]  Axel Klar,et al.  Partial Moment Entropy Approximation to Radiative Heat Transfer , 2005 .

[74]  J. McKinney General relativistic force-free electrodynamics: a new code and applications to black hole magnetospheres , 2006, astro-ph/0601410.

[75]  Light Curves from an MHD Simulation of a Black Hole Accretion Disk , 2006, astro-ph/0606615.

[76]  Richard I. Klein,et al.  Equations and Algorithms for Mixed-frame Flux-limited Diffusion Radiation Hydrodynamics , 2006 .

[77]  O. Blaes,et al.  Surface Structure in an Accretion Disk Annulus with Comparable Radiation and Gas Pressure , 2007, 0705.0314.

[78]  Mohammed Seaïd,et al.  A consistent approach for the coupling of radiation and hydrodynamics at low Mach number , 2007, J. Comput. Phys..

[79]  O. Zanotti,et al.  ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics , 2007, 0704.3206.

[80]  S. Mineshige,et al.  Why Is Supercritical Disk Accretion Feasible? , 2007, 0710.2941.

[81]  R. Teyssier,et al.  A radiative transfer scheme for cosmological reionization based on a local Eddington tensor , 2007, 0709.1544.

[82]  Jonathan C. McKinney,et al.  WHAM : a WENO-based general relativistic numerical scheme -I. Hydrodynamics , 2007, 0704.2608.

[83]  P. Huynh,et al.  HERACLES: a three-dimensional radiation hydrodynamics code , 2007 .

[84]  Aya Kubota,et al.  Modelling the behaviour of accretion flows in X-ray binaries , 2007, 0708.0148.

[85]  O. Blaes,et al.  Thermodynamics of an Accretion Disk Annulus with Comparable Radiation and Gas Pressure , 2007, 0705.0305.

[86]  Feedback from Supercritical Disk Accretion Flows: Two-dimensional Radiation-Hydrodynamic Simulations of Stable and Unstable Disks with Radiatively Driven Outflows , 2007, astro-ph/0703103.

[87]  O. Blaes,et al.  Global General Relativistic Magnetohydrodynamic Simulation of a Tilted Black Hole Accretion Disk , 2007, 0706.4303.

[88]  A. Mignone,et al.  Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index , 2007, 0704.1679.

[89]  Kris Beckwith,et al.  The Influence of Magnetic Field Geometry on the Evolution of Black Hole Accretion Flows: Similar Disks, Drastically Different Jets , 2007, 0709.3833.

[90]  A. Fabian,et al.  Broad Iron-Kα Emission Lines as a Diagnostic of Black Hole Spin , 2007, 0711.4158.

[91]  S. Shapiro,et al.  Relativistic Radiation Magnetohydrodynamics in Dynamical Spacetimes , 2008, 0802.3210.

[92]  M. C. Miller,et al.  THE TIME VARIABILITY OF GEOMETRICALLY THIN BLACK HOLE ACCRETION DISKS. I. THE SEARCH FOR MODES IN SIMULATED DISKS , 2008, 0805.2950.

[93]  J. Krolik,et al.  Where is the radiation edge in magnetized black hole accretion discs , 2008, 0801.2974.

[94]  A. Tchekhovskoy,et al.  Three-Dimensional Simulations of Magnetized Thin Accretion Disks around Black Holes: Stress in the Plunging Region , 2008, 0808.2860.

[95]  Marta Volonteri,et al.  Cosmological Black Hole Spin Evolution by Mergers and Accretion , 2008, 0802.0025.

[96]  Thomas Christen,et al.  Minimum entropy production closure of the photo-hydrodynamic equations for radiative heat transfer , 2008, 0812.3321.

[97]  A. Sa̧dowski SLIM DISKS AROUND KERR BLACK HOLES REVISITED , 2009, 0906.0355.

[98]  O. Blaes,et al.  RADIATION-DOMINATED DISKS ARE THERMALLY STABLE , 2008, 0809.1708.

[99]  O. Blaes,et al.  TURBULENT STRESSES IN LOCAL SIMULATIONS OF RADIATION-DOMINATED ACCRETION DISKS, AND THE POSSIBILITY OF THE LIGHTMAN–EARDLEY INSTABILITY , 2009, 0908.1117.

[100]  S. Mineshige,et al.  Modified Slim-Disk Model Based on Radiation-Hydrodynamic Simulation Data: The Conflict Between Outflow and Photon Trapping , 2009, 0904.4598.

[101]  Oscar Reula,et al.  Beyond ideal MHD: towards a more realistic modelling of relativistic astrophysical plasmas , 2008, 0810.1838.

[102]  D. Meier,et al.  GENERAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATIONS OF THE HARD STATE AS A MAGNETICALLY DOMINATED ACCRETION FLOW , 2008, 0810.1082.

[103]  J. Krolik,et al.  TRANSPORT OF LARGE-SCALE POLOIDAL FLUX IN BLACK HOLE ACCRETION , 2009, 0906.2784.

[104]  J. Krolik,et al.  DIRECT CALCULATION OF THE RADIATIVE EFFICIENCY OF AN ACCRETION DISK AROUND A BLACK HOLE , 2008, 0808.3140.

[105]  R. Blandford,et al.  Stability of relativistic jets from rotating, accreting black holes via fully three-dimensional magnetohydrodynamic simulations , 2008, 0812.1060.

[106]  Giovanni Russo,et al.  On a Class of Uniformly Accurate IMEX Runge--Kutta Schemes and Applications to Hyperbolic Systems with Relaxation , 2009, SIAM J. Sci. Comput..

[107]  R. McLure,et al.  THE CANADA–FRANCE HIGH-z QUASAR SURVEY: NINE NEW QUASARS AND THE LUMINOSITY FUNCTION AT REDSHIFT 6 , 2009, 0912.0281.

[108]  P. Armitage,et al.  CONNECTIONS BETWEEN LOCAL AND GLOBAL TURBULENCE IN ACCRETION DISKS , 2010, 1002.3611.

[109]  Stephen R. Green,et al.  Numerical parameter survey of non‐radiative black hole accretion: flow structure and variability of the rotation measure , 2010, 1011.5498.

[110]  A. Tchekhovskoy,et al.  Simulations of magnetized discs around black holes: Effects of black hole spin, disc thickness and magnetic field geometry , 2010, 1003.0966.

[111]  J. Krolik,et al.  DEPENDENCE OF INNER ACCRETION DISK STRESS ON PARAMETERS: THE SCHWARZSCHILD CASE , 2010, 1001.4809.

[112]  S. Mineshige,et al.  A Novel Jet Model: Magnetically Collimated, Radiation-Pressure Driven Jet , 2010, 1009.0161.

[113]  O. Blaes,et al.  DISSIPATION AND VERTICAL ENERGY TRANSPORT IN RADIATION-DOMINATED ACCRETION DISKS , 2011, 1103.5052.

[114]  M. Begelman Radiatively inefficient accretion: breezes, winds and hyperaccretion , 2011, 1110.5356.

[115]  J. Krolik,et al.  RADIATIVE EFFICIENCY AND THERMAL SPECTRUM OF ACCRETION ONTO SCHWARZSCHILD BLACK HOLES , 2011, 1105.2825.

[116]  Nathaniel R. Butler,et al.  A Possible Relativistic Jetted Outburst from a Massive Black Hole Fed by a Tidally Disrupted Star , 2011, Science.

[117]  L. Rezzolla,et al.  General relativistic radiation hydrodynamics of accretion flows - I. Bondi-Hoyle accretion , 2011, 1105.5615.

[118]  Harvard,et al.  Efficient Generation of Jets from Magnetically Arrested Accretion on a Rapidly Spinning Black Hole , 2011, 1108.0412.

[119]  Nozomu Kawakatu,et al.  New method for exploring super-Eddington active galactic nuclei by near-infrared observations , 2011, 1107.2185.

[120]  Cory D. Hauck,et al.  High-Order Entropy-Based Closures for Linear Transport in Slab Geometries , 2011 .

[121]  C. Gammie,et al.  PAIR PRODUCTION IN LOW-LUMINOSITY GALACTIC NUCLEI , 2011, 1104.2042.

[122]  S. Mineshige,et al.  GLOBAL STRUCTURE OF THREE DISTINCT ACCRETION FLOWS AND OUTFLOWS AROUND BLACK HOLES FROM TWO-DIMENSIONAL RADIATION-MAGNETOHYDRODYNAMIC SIMULATIONS , 2011, 1105.5474.

[123]  G. Chabrier,et al.  Physical and radiative properties of the first-core accretion shock , 2011, 1102.2921.

[124]  Ryan Chornock,et al.  Birth of a relativistic outflow in the unusual γ-ray transient Swift J164449.3+573451 , 2011, Nature.

[125]  E. O. Ofek,et al.  An Extremely Luminous Panchromatic Outburst from the Nucleus of a Distant Galaxy , 2011, Science.

[126]  Richard G. McMahon,et al.  A luminous quasar at a redshift of z = 7.085 , 2011, Nature.

[127]  P. Giommi,et al.  Relativistic jet activity from the tidal disruption of a star by a massive black hole , 2011, Nature.

[128]  P. Armitage,et al.  Turbulence in global simulations of magnetized thin accretion discs , 2011, 1105.1789.

[129]  Olindo Zanotti,et al.  General relativistic radiation hydrodynamics of accretion flows – II. Treating stiff source terms and exploring physical limitations , 2012, 1206.6662.

[130]  Princeton,et al.  General relativistic magnetohydrodynamic simulations of magnetically choked accretion flows around black holes , 2012, 1201.4163.

[131]  James M. Stone,et al.  A GODUNOV METHOD FOR MULTIDIMENSIONAL RADIATION MAGNETOHYDRODYNAMICS BASED ON A VARIABLE EDDINGTON TENSOR , 2012, 1201.2223.

[132]  Alan E. E. Rogers,et al.  Jet-Launching Structure Resolved Near the Supermassive Black Hole in M87 , 2012, Science.

[133]  M. Begelman FORCE-FEEDING BLACK HOLES , 2012, 1203.1628.

[134]  R. Narayan,et al.  GRMHD simulations of magnetized advection‐dominated accretion on a non‐spinning black hole: role of outflows , 2012, 1206.1213.

[135]  Stanford,et al.  Prograde and retrograde black holes: whose jet is more powerful? , 2012, 1201.4385.

[136]  Andrew C. Fabian,et al.  Observational Evidence of Active Galactic Nuclei Feedback , 2012 .

[137]  P. Anninos,et al.  NUMERICAL SIMULATIONS OF OPTICALLY THICK ACCRETION ONTO A BLACK HOLE. I. SPHERICAL CASE , 2012, 1204.5538.

[138]  J. Dexter,et al.  GRMHD simulations of accretion onto Sgr A*: How important are radiative losses? , 2012, 1206.3976.

[139]  Takashi Yoshida,et al.  COMPTONIZED PHOTON SPECTRA OF SUPERCRITICAL BLACK HOLE ACCRETION FLOWS WITH APPLICATION TO ULTRALUMINOUS X-RAY SOURCES , 2012 .

[140]  A. Tchekhovskoy,et al.  General Relativistic Modeling of Magnetized Jets from Accreting Black Holes , 2012, 1202.2864.

[141]  R. Penna,et al.  SAGITTARIUS A* ACCRETION FLOW AND BLACK HOLE PARAMETERS FROM GENERAL RELATIVISTIC DYNAMICAL AND POLARIZED RADIATIVE MODELING , 2010, 1007.4832.

[142]  K. Kotake,et al.  FULLY GENERAL RELATIVISTIC SIMULATIONS OF CORE-COLLAPSE SUPERNOVAE WITH AN APPROXIMATE NEUTRINO TRANSPORT , 2012, 1202.2487.

[143]  Y. Sekiguchi,et al.  Radiation magnetohydrodynamics for black hole-torus system in full general relativity: A step toward physical simulation , 2012, 1206.5911.

[144]  S. Abbassi,et al.  Radiation spectrum of a magnetized supercritical accretion disc with thermal conduction , 2013, 1304.6922.

[145]  E. Ostriker,et al.  A TWO-MOMENT RADIATION HYDRODYNAMICS MODULE IN ATHENA USING A TIME-EXPLICIT GODUNOV METHOD , 2013, 1306.0010.

[146]  Mareki Honma,et al.  THE INNERMOST COLLIMATION STRUCTURE OF THE M87 JET DOWN TO ∼10 SCHWARZSCHILD RADII , 2013, 1308.1411.

[147]  R. Narayan,et al.  Energy, momentum and mass outflows and feedback from thick accretion discs around rotating black holes , 2013, 1307.1143.

[148]  J. Stone,et al.  ON THE THERMAL STABILITY OF RADIATION-DOMINATED ACCRETION DISKS , 2013, 1309.5646.

[149]  Yuji Kanno,et al.  Kinetic Scheme for Solving the M1 Model of Radiative Transfer , 2013 .

[150]  R. Blandford,et al.  Alignment of Magnetized Accretion Disks and Relativistic Jets with Spinning Black Holes , 2012, Science.

[151]  S. Mineshige,et al.  Clumpy Outflows from Supercritical Accretion Flow , 2013, 1305.1023.

[152]  Hiroyuki R. Takahashi,et al.  A Numerical Treatment of Anisotropic Radiation Fields Coupled with Relativistic Resistive Magnetofluids , 2013 .

[153]  Z. Younsi,et al.  Covariant Compton scattering kernel in general relativistic radiative transfer , 2013, 1305.6059.

[154]  Hiroyuki R. Takahashi,et al.  EXPLICIT–IMPLICIT SCHEME FOR RELATIVISTIC RADIATION HYDRODYNAMICS , 2012, 1212.4910.

[155]  A. Tchekhovskoy,et al.  Semi-implicit scheme for treating radiation under M1 closure in general relativistic conservative fluid dynamics codes , 2012, 1212.5050.

[156]  Catalin Trenchea,et al.  Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows , 2014, Appl. Math. Lett..

[157]  Columbia,et al.  Swift J1644+57 gone MAD: the case for dynamically-important magnetic flux threading the black hole in a jetted tidal disruption event , 2013, 1301.1982.

[158]  Inga Kamp,et al.  European Physical Journal Web of Conferences , 2015 .