3D GRMHD code: RAISHIN Yosuke Mizuno Institute for Theoretical Physics Goethe University Frankfurt am Main BH Cam code-comparison workshop, January 20-21, 2015

Code General Information Mizuno et al. 2006a, 2011c, & progress RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D ideal GRMHD equations (metric is static) Program Language: Fortran 90 Multi-dimension (1D, 2D, 3D) Special & General relativity (static metric), three different code package Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist coordinates and Kerr-Schild coordinates) Different schemes of numerical accuracy for numerical model (spatial reconstruction, approximate Riemann solver, constrained transport schemes, time advance, & inversion) Using constant G-law, polytropic, and approximate Equation of State (Synge-type) Uniform & non-uniform grid Parallel computing (based on MPI)

Basic Equations (SRMHD) Conserved Form Primitive variables Conserved variables Flux Magnetic field 4-vector (Magnetic field measured in comoving frame)

Basic Equations (GRMHD) Metric Conserved Form a: lapse function, bi: shift vector Primitive variables Conserved variables Flux Source term Magnetic field 4-vector Energy-momentum tensor

Basic Equations (GRMHD) Lorentz factor

Flow Chart for Calculation Reconstruction (Pn : cell-center to cell-surface) 2. Calculation of Flux at cell-surface 3. Integrate hyperbolic equations => Un+1 4. Convert from Un+1 to Pn+1 Primitive Variables: P Conserved Variables: U Flux: F Source: S Fi-1/2 Ui Fi+1/2 Pi-1 Pi Pi+1 Si

Reconstruction Cell-centered variables (Pi) → right and left side of Cell-interface variables(PLi+1/2, PRi+1/2) Minmod & MC Flux-limiter (Piecewise linear Method) 2nd order at smooth region Convex ENO (Liu & Osher 1998) 3rd order at smooth region Piecewise Parabolic Method (Marti & Muller 1996) 4th order at smooth region Weighted ENO, WENO-Z, WENO-M (Jiang & Shu 1996; Borges et al. 2008) 5th order at smooth region Monotonicity Preserving (Suresh & Huynh 1997) MPWENO5 (Balsara & Shu 2000) Logarithmic 3rd order limiter (Cada & Torrilhon 2009) Piecewise linear interpolation PLi+1/2 PRi+1/2 Pni-1 Pni Pni+1

Approximate Riemann Solver Calculate numerical flux at cell-inteface from reconstructed cell-interface variables based on Riemann problem We use HLL approximate Riemann solver Need only the maximum left- and right- going wave speeds (in MHD case, fast magnetosonic mode) lR, lL: fastest characteristic speed HLL flux lL t lR M Fi-1/2 Ui Fi+1/2 L R x Pi-1 Pi Pi+1 If lL >0 FHLL=FL lL < 0 < lR , FHLL=FM lR < 0 FHLL=FR

Approximate Riemann Solver HLL Approximate Riemann solver: single state in Riemann fan HLLC Approximate Riemann solver: two-state in Riemann fan (Mignone & Bodo 2006, Honkkila & Janhunen 2007) (for SRMHD only) HLLD Approximate Riemann solver: six-state in Riemann fan (Mignone et al. 2009) (for SRMHD only) Roe-type full wave decomposition Riemann solver (Anton et al. 2010)

Wave speed To calculate numerical flux at each cell-boundary via Riemann solver, we need to know wave speed in each directions lE=vi, entropy wave Alfven waves Magneto-acoustic waves are found from the quartic equation Some simple estimation for fast magnetosonic wave => Leismann et al. (2005), no numerical iteration

Constrained Transport Differential Equations The evolution equation can keep divergence free magnetic field If treat the induction equation as all other conservation laws, it can not maintain divergence free magnetic field → We need spatial treatment for magnetic field evolution Constrained transport scheme Evans & Hawley’s Constrained Transport (need staggered mesh) Toth’s constrained transport (flux-CT) (Toth 2000) for SRMHD & GRMHD Fixed Flux-CT, Upwind Flux-CT (Gardiner & Stone 2005, 2007), for SRMHD only Other method Diffusive cleaning (GLM formulation) for RRMHD only

Time evolution System of Conservation Equations We use multistep TVD Runge-Kutta method for time advance of conservation equations (RK2: 2nd-order, RK3: 3rd-order in time) RK2, RK3: first step RK2: second step (a=2, b=1) RK3: second and third step (a=4, b=3)

Recovery step The GRMHD code require a calculation of primitive variables from conservative variables. The forward transformation (primitive → conserved) has a close-form solution, but the inverse transformation (conserved → primitive) requires the solution of a set of five nonlinear equations Method Noble’s 2D method (Noble et al. 2005) Mignone & McKinney’s method (Mignone & McKinney 2007)

Variable EoS Mignone & McKinney 2007 In the theory of relativistic perfect gases, specific enthalpy is a function of temperature alone (Synge 1957) Q: temperature p/r K2, K3: the order 2 and 3 of modified Bessel functions Constant G-law EoS: G: constant specific heat ratio Taub’s fundamental inequality (Taub 1948) Q → 0, Geq → 5/3, Q → ∞, Geq → 4/3 TM approximate EoS (Mathew 1971, Mignone et al. 2005)

General (Approximate) EoS Mignone & McKinney 2007 In the theory of relativistic perfect single gases, specific enthalpy is a function of temperature alone (Synge 1957) Q: temperature p/r K2, K3: the order 2 and 3 of modified Bessel functions Constant G-law EoS (ideal EoS) : G: constant specific heat ratio Taub’s fundamental inequality(Taub 1948) Q → 0, Geq → 5/3, Q → ∞, Geq → 4/3 Solid: Synge EoS Dotted: ideal + G=5/3 Dashed: ideal+ G=4/3 Dash-dotted: TM EoS TM EoS (approximate Synge’s EoS) (Mignone et al. 2005) c/sqrt(3)

Numerical Tests

Code Accuracy (L1 norm) 1D CP Alfven wave propagation test L1 norm errors of magnetic field vy shows almost 2nd order accuracy

Code Accuracy (grid number vs computer time) 1D shock-tube (Balsara Test 1) with 1 CPU calculated to t=0.4 tsim ∝ Nx2 Number of grid

Parallelization Accuracy 1D shock-tube (Balsara Test 1) in 3D Cartesian box, calculated to t=0.4 99% 98% 90% T(1) / T(N) Number of CPU

1D Relativistic MHD Shock-Tube Balsara Test1 (Balsara 2001) Mizuno et al. 2006 The results show good agreement of the exact solution calculated by Giacommazo & Rezzolla (2006). Minmod slope-limiter and CENO reconstructions are more diffusive than the MC slope-limiter and PPM reconstructions. Although MC slope limiter and PPM reconstructions can resolve the discontinuities sharply, some small oscillations are seen at the discontinuities. FR SR SS FR CD Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM 400 computational zones

Advection of Magnetic Field Loop No advection Advection of a weak magnetic field loop in an uniform velocity field 2D: (vx, vy)=(0.6,0.3) 3D: (vx,vy,vz)=(0.3,0.3,0.6) Periodic boundary in all direction Run until return to initial position in advection case B2 Advection Volume-averaged magnetic energy density (2D) 3D B2 Nx=512 256 128 Advection No advection

Cylindrical Explosion Cylindrical explosion in magnetized medium (propagation of strong shock waves) Pc=1.0, rc=0.01 (R<1.0) Pe=5x10-4, re=10-4 Bx=0.1 (uniform)

1D Bondi Accretion ur vr 1D Bondi flow with radial B-field (b=1) G=4/3, rc=8rg r/rg Results follow initial Bondi flow structure ur vr r/rg r/rg

2D torus (Hydro) 2D geometrically thick torus (Fishbone & Moncrief 1976) with no magnetic field. a/M=0, Kerr-Schild coordinates r

Resistive Relativistic MHD

Relativistic Regime Kinetic energy >> rest-mass energy Fluid velocity ~ light speed Lorentz factor g>> 1 Relativistic jets/ejecta/wind/blast waves (shocks) in AGNs, GRBs, Pulsars Thermal energy >> rest-mass energy Plasma temperature >> ion rest mass energy p/r c2 ~ kBT/mc2 >> 1 GRBs, magnetar flare?, Pulsar wind nebulae Magnetic energy >> rest-mass energy Magnetization parameter s >> 1 s = Poyniting to kinetic energy ratio = B2/4pr c2g2 Pulsars magnetosphere, Magnetars Gravitational energy >> rest-mass energy GMm/rmc2 = rg/r > 1 Black hole, Neutron star Radiation energy >> rest-mass energy E’r /rc2 >>1 Supercritical accretion flow

Relativistic Jets Radio observation of M87 jet Relativistic jets: outflow of highly collimated plasma Microquasars, Active Galactic Nuclei, Gamma-Ray Bursts, Jet velocity ～c Generic systems: Compact object（White Dwarf, Neutron Star, Black Hole）+ Accretion Disk Key Issues of Relativistic Jets Acceleration & Collimation Propagation & Stability Modeling for Jet Production Magnetohydrodynamics (MHD) Relativity (SR or GR) Modeling of Jet Emission Particle Acceleration Radiation mechanism

Relativistic Jets in Universe Mirabel & Rodoriguez 1998

Ultra-Fast TeV Flare in Blazars Ultra-Fast TeV flares are observed in some Blazars. Vary on timescale as sort as tv~3min << Rs/c ~ 3M9 hour For the TeV emission to escape pair creation Γem>50 is required (Begelman, Fabian & Rees 2008) But PKS 2155-304, Mrk 501 show “moderately” superluminal ejections (vapp ~several c) Emitter must be compact and extremely fast Model for the Fast TeV flaring Internal: Magnetic Reconnection inside jet (Giannios et al. 2009) External: Recollimation shock (Bromberg & Levinson 2009) PKS2155-304 (Aharonian et al. 2007) See also Mrk501, PKS1222+21 Giannios et al.(2009)

Magnetic Reconnection in Relativistic Astrophysical Objects Pulsar Magnetosphere & Striped pulsar wind obliquely rotating magnetosphere forms stripes of opposite magnetic polarity in equatorial belt magnetic dissipation via magnetic reconnection would be main energy conversion mechanism Magnetar Flares May be triggered by magnetic reconnection at equatorial current sheet Spitkovsky (2006)

Purpose of Study Quite often numerical simulations using ideal RMHD exhibit violent magnetic reconnection. The magnetic reconnection observed in ideal RMHD simulations is due to purely numerical resistivity, occurs as a result of truncation errors Fully depends on the numerical scheme and resolution. Therefore, to allow the control of magnetic reconnection according to a physical model of resistivity, numerical codes solving the resistive RMHD (RRMHD) equations are highly desirable. We have newly developed RRMHD code and investigated the role of the equation of state in RRMHD regime.

Applicability of Hydrodynamic Approximation To apply hydrodynamic approximation, we need the condition: Spatial scale >> mean free path Time scale >> collision time These are not necessarily satisfied in many astrophysical plasmas E.g., solar corona, galactic halo, cluster of galaxies etc. But in plasmas with magnetic field, the effective mean free path is given by the ion Larmor radius. Hence if the size of phenomenon is much larger than the ion Larmor radius, hydrodynamic approximation can be used.

Applicability of MHD Approximation MHD describe macroscopic behavior of plasmas if Spatial scale >> ion Larmor radius Time scale >> ion Larmor period But MHD can not treat Particle acceleration Origin of resistivity Electromagnetic waves

Ideal / Resistive RMHD Eqs Ideal RMHD Resistive RMHD Solve 11 equations (8 in ideal MHD) Need a closure relation between J and E => Ohm’s law

Ohm’s law Relativistic Ohm’s law (Blackman & Field 1993 etc.) isotropic diffusion in comoving frame (most simple one) Lorentz transformation in lab frame Relativistic Ohm’s law with istoropic diffusion ideal MHD limit (conductivity: s => infinity) Charge current disappear in the Ohm’s law (degeneracy of equations, EM wave is decupled)

Difference in Ideal & Resistive RMHD Evolution of EM fields ideal resistive unnecessary to solve Ampere’s law in ideal MHD E field can be determined directly from Ohm’s law with Ohm’s law 2 additional equations should be solved Disadvantage of RRMHD Courant condition ∇・E=4πq should be satisfied as well as ∇・B=0

Basic Equations for Ideal / Resistive RMHD Ideal RMHD Hyperbolic equations Source term Stiff term

Numerical Integration Resistive RMHD Constraint Hyperbolic equations Solve Relativistic Resistive MHD equations by taking care of 1. stiff equations appeared in Ampere’s law 2. constraints ( no monopole, Gauss’s law) 3. Courant conditions (the largest characteristic wave speed is always light speed.) Source term Stiff term

For Numerical Simulations Basic Equations for RRMHD Physical quantities Primitive Variables Conserved Variables Flux Source term Operator splitting (Strang’s method) to divide for stiff term

Basics of Numerical RMHD Code Non-conservative form (De Villier & Hawley (2003), Anninos et al.(2005)) U=U(P) - conserved variables, P – primitive variables F- numerical flux of U where Merit: they solve the internal energy equation rather than energy equation. → advantage in regions where the internal energy small compared to total energy (such as supersonic flow) Recover of primitive variables are fairly straightforward Demerit: It can not applied high resolution shock-capturing method and artificial viscosity must be used for handling discontinuities

Basics of Numerical RMHD Code Conservative form System of Conservation Equations U=U(P) - conserved variables, P – primitive variables F- numerical flux of U, S - source of U Merit: Numerically well maintain conserved variables High resolution shock-capturing method (Godonuv scheme) can be applied to RMHD equations Demerit: These schemes must recover primitive variables P by numerically solving the system of equations after each step (because the schemes evolve conservative variables U)

Finite Difference (Volume) Method Conservative form of wave equation flux Finite difference FTCS scheme Upwind scheme Lax-Wendroff scheme

Difficulty of Handling Shock Wave Numerical oscillation (overshoot) Diffuse shock surface Time evolution of wave equation with discontinuity using Lax-Wendroff scheme (2nd order) initial In numerical hydrodynamic simulations, we need sharp shock structure (less diffusivity around discontinuity) no numerical oscillation around discontinuity higher-order resolution at smooth region handling extreme case (strong shock, strong magnetic field, high Lorentz factor) Divergence-free magnetic field (MHD)

Flow Chart for Calculation Reconstruction (Pn : cell-center to cell-surface) 2. Calculation of Flux at cell-surface 3. Integrate hyperbolic equations => Un+1 4. Integrate stiff term (E field) 5. Convert from Un+1 to Pn+1 Primitive Variables Conserved Variables Flux Fi-1/2 Ui Fi+1/2 Pi-1 Pi Pi+1

Difficulty of RRMHD 1. Constraint should be satisfied both constraint numerically 2. Ampere’s law Equation becomes stiff at high conductivity

Constraints Approaching Divergence cleaning method (Dedner et al. 2002, Komissarov 2007) Introduce additional field F & Y (for numerical noise) advect & decay in time

Stiff Equation Komissarov (2007) Problem comes from difference between dynamical time scale and diffusive time scale => analytical solution Ampere’s law diffusion (stiff) term Operator splitting method Hyperbolic + source term Solve by HLL method Analytical solution source term (stiff part) Solve (ordinary differential) eqaution

Time Evolution (ideal RMHD) System of Conservation Equations We use multistep TVD Runge-Kutta method for time advance of conservation equations (RK2: 2nd-order, RK3: 3rd-order in time) RK2, RK3: first step RK2: second step (a=2, b=1) RK3: second and third step (a=4, b=3)

Flow Chart for Calculation (RRMHD) Strang Splitting Method Step1: integrate diffusion term in half-time step Step2: integrate advection term in half-time step Un=(En+1/2, Bn) Step3: integrate advection term in full-time step Step4: integrate diffusion term in full-time step (En+1, Bn+1)=Un+1

Recovery step The GRMHD code require a calculation of primitive variables from conservative variables. The forward transformation (primitive → conserved) has a close-form solution, but the inverse transformation (conserved → primitive) requires the solution of a set of five nonlinear equations Method Noble’s 2D method (Noble et al. 2005) Mignone & McKinney’s method (Mignone & McKinney 2007)

Noble’s 2D method Conserved quantities(D,S,t,B) → primitive variables (r,p,v,B) Solve two-algebraic equations for two independent variables W≡hg2 and v2 by using 2-variable Newton-Raphson iteration method W and v2 →primitive variables r p, and v Mignone & McKinney (2007): Implemented from Noble’s method for variable EoS

General (Approximate) EoS Mignone & McKinney 2007 In the theory of relativistic perfect single gases, specific enthalpy is a function of temperature alone (Synge 1957) Q: temperature p/r K2, K3: the order 2 and 3 of modified Bessel functions Constant G-law EoS (ideal EoS) : G: constant specific heat ratio Taub’s fundamental inequality(Taub 1948) Q → 0, Geq → 5/3, Q → ∞, Geq → 4/3 Solid: Synge EoS Dotted: ideal + G=5/3 Dashed: ideal+ G=4/3 Dash-dotted: TM EoS TM EoS (approximate Synge’s EoS) (Mignone et al. 2005) c/sqrt(3)

2. 1. Numerical Tests ideal RMHD

1D Relativistic MHD Shock-Tube Exact solution: Giacomazzo & Rezzolla (2006)

1D Relativistic MHD Shock-Tube Balsara Test1 (Balsara 2001) Mizuno et al. 2006 The results show good agreement of the exact solution calculated by Giacommazo & Rezzolla (2006). Minmod slope-limiter and CENO reconstructions are more diffusive than the MC slope-limiter and PPM reconstructions. Although MC slope limiter and PPM reconstructions can resolve the discontinuities sharply, some small oscillations are seen at the discontinuities. FR SR SS FR CD Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM 400 computational zones

Advection of Magnetic Field Loop No advection Advection of a weak magnetic field loop in an uniform velocity field 2D: (vx, vy)=(0.6,0.3) 3D: (vx,vy,vz)=(0.3,0.3,0.6) Periodic boundary in all direction Run until return to initial position in advection case B2 Advection Volume-averaged magnetic energy density (2D) 3D B2 Nx=512 256 128 Advection No advection

Cylindrical Explosion Cylindrical explosion in magnetized medium (propagation of strong shock waves) Pc=1.0, rc=0.01 (R<1.0) Pe=5x10-4, re=10-4 Bx=0.1 (uniform)

Numerical Tests

1D CP Alfven wave propagation test Aim: Recover of ideal RMHD regime in high conductivity Propagation of large amplitude circular-polarized Alfven wave along uniform magnetic field Exact solution: Del Zanna et al.(2007) in ideal RMHD limit Bx=B０, vx=0, k: wave number, zA: amplitude of wave =p=1, B0=0.46188 => vA=0.25c, ideal EoS with G=2 Using high conductivity s=105

1D CP Alfven wave propagation test Numerical results at t=4 (one Alfven crossing time) Solid: exact solution Dotted: Nx=50 Dashed: Nx=100 Dash-dotted: Nx=200 New RRMHD code reproduces ideal RMHD solution when conductivity is high L1 norm errors of magnetic field By almost 2nd order accuracy

1D Self-Similar Current Sheet Test Assumption: Magnetic pressure << gas pressure Magnetic field configuration: B=[0, By(x,t),0], By(x,t) changes sign within a thin current layer (thickness Dl) The evolution of thin current layer is a slow diffusive expansion due to resistivity and described by diffusion equations As the thickness of the layer becomes much larger than Dl, the expansion becomes self-similar: c=t/x2, erf: error function Test simulation: Initial solution at t=1 with p=50, r=1, E=v=0, s=100 with G=2

1D Self-Similar Current Sheet Test dotted & dashed: analytical solution at t=1 & 10 Solid: numerical solution at t=10 Numerical Simulation shows good agreement with exact solutions with moderate conductivity regime

1D Shock-Tube Test (Brio & Wu) Aim: Check the effect of resistivity (conductivity) Simple MHD version of Brio & Wu test (rL, pL, ByL) = (1, 1, 0.5), (rR, pR, ByR)=(0.125, 0.1, -0.5) Ideal EoS with G=2 Orange solid: s=0 Green dash-two-dotted: s=10 Red dash-dotted: s=102 Purple dashed: s=103 Blue dotted: s=105 Black solid: exact solution in ideal RMHD Smooth change from a wave-like solution (s=0) to ideal-MHD solution (s=105)

1D Shock-Tube Test (Balsara 2) Aim: check the effect of choosing EoS in RRMHD Balsara Test 2 Using ideal EoS (G=5/3) & approximate TM EoS Changing conductivity from s=0 to 103 Mildly relativistic blast wave propagates with 1.3 < g < 1.4

1D Shock-Tube Test (Balsara 2) SS & FS CD FR SR Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: exact solution in ideal RMHD The solutions: Fast Rarefaction, Slow Rarefaction, Contact Discontinuity, Slow Shock and Fast Shock.

1D Shock-Tube Test (Balsara 2) SS & FS FR CD SR Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: exact solution in ideal RMHD The solutions are same but quantitatively different. rarefaction waves and shocks propagate with smaller velocities <= lower sound speed in TM EoSs relatives to overestimated sound speed in ideal EoS these properties are consistent with in ideal RMHD case

2D Cylindrical Explosion Cylindrical blast wave expanding into uniform magnetic field Standard test for ideal RMHD code No exact solution in multi-dimensional tests Initial Condition Density, pressure: R<0.8, p=1, r=0.01 0.8<R<1.0 decrease exponentially to ambient gas R>1.0 p=r=0.001 Magnetic field: uniform in x-direction with B=0.05 Using different conductivity s=0-105 Using ideal EoS with G=4/3 and TM EoS

Global Structure in Cylindrical Explosion qualitatively similar to ideal RMHD results No different between ideal EoS and TM EoS

1D cut of Cylindrical Explosion Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: s=105 At high conductivity (s > 103) no difference (recovers ideal MHD solution) conductivity ↓ maximum gas and mag pressure ↓ No mag pressure increase for s=0

2D Kelvin-Helmholtz Instability Linear and nonlinear growth of 2D Kelvin-Helmholtz instability (KHI) & magnetic field amplification via KHI Initial condition Shear velocity profile: Uniform gas pressure p=1.0 Density: r=1.0 in the region vsh=0.5, r=10-2 in the region vsh=-0.5 Magnetic Field: Single mode perturbation: Simulation box: -0.5 < x < 0.5, -1 < y < 1 a=0.01, characteristic thickness of shear layer vsh=0.5 => relative g=2.29 mp=0.5, mt=1.0 A0=0.1, a=0.1

Growth Rate of KHI Initial linear growth with almost same growth rate Amplitude of perturbation Volume-averaged Poloidal field Purple dash-two-dotted: s=0 Green dash-dotted: s=10 Red dashed: s=102 Blue dotted: s=103 Black solid: s=105 Initial linear growth with almost same growth rate Maximum amplitude; transition from linear to nonlinear Poloidal field amplification via stretching due to main vortex developed by KHI Larger poloidal field amplification occurs for TM EoS than for ideal EoS

2D KHI Global Structure (ideal EoS) Formation of main vortex by growth of KHI in linear growth phase secondary vortex? main vortex is distorted and stretched in nonlinear phase B-field amplified by shear in vortex in linear and stretching in nonlinear

2D KHI Global Structure (TM EoS) Formation of main vortex by growth of KHI in linear growth phase no secondary vortex main vortex is distorted and stretched in nonlinear phase vortex becomes strongly elongated in nonlinear phase Created structure is very different in ideal and TM EoSs

Field Amplification in KHI Field amplification structure for different conductivities Conductivity low, magnetic field amplification is weaker Field amplification is a result of fluid motion in the vortex B-field follows fluid motion, like ideal MHD, strongly twisted in high conductivity conductivity decline, B-field is no longer strongly coupled to the fluid motion Therefore B-field is not strongly twisted

Relativistic Magnetic Reconnection Assumption Consider Pestchek-type magnetic reconnection Initial condition Harris-type model（anti-parallel magnetic field） Anomalous resistivity for triggering magnetic reconnection （r<0.8） Results B-filed：typical X-type topology Density：Plasmoid Reconnection outflow: ~0.8c

2D Relativistic Magnetic Reconnection Consider Pestchek-type reconnection Initial condition: Harris-like model Uniform density & gas pressure outside current sheet, rb=pb=0.1 Density & gas pressure: Magnetic field: Current: Resistivity (anomalous resistivity in r<rh): hb=1/sb=10-3, h0=1.0, rh=0.8 Electric field:

Global Structure of Relativistic MR Plasmoid t=100 Slow shock Strong current flow

Time Evolution of Relativistic MR Outflow gradually accelerates and saturates t~60 with vx~0.8c TM EoS case slightly faster than ideal EoS case Magnetic energy converted to thermal and kinetic energies (acceleration of outflow) TM EoS case has larger reconnection rate than ideal EoS. Different EoSs lead to a quantitative difference in relativistic magnetic reconnection Reconnection outflow speed Solid: ideal EoS Dashed: TM EoS Magnetic energy Reconnection rate time

Summary of RAISHIN code RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D ideal GRMHD equations (metric is static) Program Language: Fortran 90 Multi-dimension (1D, 2D, 3D) Special & General relativity (static metric), three different code package (SRMHD, GRMHD, RRMHD) Different coordinates (SRMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist coordinates and Kerr-Schild coordinates) Different schemes is applied in each steps Using constant G-law, polytropic, and approximate Equation of State (Synge-type) Parallel computing (based on MPI)