3D GRMHD code: RAISHIN

Yosuke MizunoInstitute for Theoretical Physics

Goethe University Frankfurt am Main

BH Cam code-comparison workshop, January 20-21, 2015

Code General Information• 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 -law, polytropic, and approximate Equation of State (Synge-type)

• Uniform & non-uniform grid• Parallel computing (based on MPI)

Mizuno et al. 2006a, 2011c, & progress

Basic Equations (SRMHD)Conserved Form

Primitive variables Conserved variables Flux

Magnetic field 4-vector (Magnetic field measured in comoving frame)

Energy-momentum tensor

Basic Equations (GRMHD)Conserved Form

Primitive variables Conserved variables Flux

Source term

Metric

Magnetic field 4-vector

: lapse function, i: shift vector

Flow Chart for Calculation

Primitive Variables: P

Conserved Variables: U

Flux: F

Pi Pi+1Pi-1

Fi-1/2 Fi+1/2Ui

1. 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

Source: S

Si

ReconstructionCell-centered variables (Pi)→ right and left side of Cell-interface variables(PL

i+1/2, PRi+1/2)

PLi+1/2 PR

i+1/2

Piecewise linear interpolation

• 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)

• 5th order at smooth region• MPWENO5 (Balsara & Shu 2000)• Logarithmic 3rd order limiter (Cada & Torrilhon 2009)

Pni

Pni+1

Pni-1

Approximate Riemann Solver

Pi Pi+1Pi-1

Fi-1/2 Fi+1/2Ui

x

t

HLL flux

RL: fastest characteristic speed

RL

If L >0 FHLL=FL

L < 0 < R , FHLL=FM

R < 0 FHLL=FR

L R

M

• 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)

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

E=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- 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

Differential Equations

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 (=2, =1)

RK3: second and third step (=4, =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)

General (Approximate) EoS• In the theory of relativistic perfect single gases, specific enthalpy is a function of temperature alone (Synge 1957)

temperature p/K2, K3: the order 2 and 3 of modified Bessel functions

• Constant -law EoS (ideal EoS) :• : constant specific heat ratio

• Taub’s fundamental inequality(Taub 1948)

→ 0, eq → 5/3, → ∞, eq → 4/3

• TM EoS (approximate Synge’s EoS) (Mignone et al. 2005)

Mignone & McKinney 2007

Solid: Synge EoSDotted: ideal + =5/3Dashed: ideal+ =4/3Dash-dotted: TM EoS

c/sqrt(3)

Numerical Tests

L1 norm errors of magnetic field vy shows almost 2nd order accuracy

Code Accuracy (L1 norm)1D CP Alfven wave propagation test

Code Accuracy (grid number vs computer time)

1D shock-tube (Balsara Test 1) with 1 CPU calculated to t=0.4

Number of grid

tsim N∝ x2

Parallelization Accuracy

90%

98%

99%

1D shock-tube (Balsara Test 1) in 3D Cartesian box, calculated to t=0.4

Number of CPU

T(1) / T(N)

1D Relativistic MHD Shock-TubeBalsara Test1 (Balsara 2001)

Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM

• 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.

400 computational zones

FR

FR

SR

CD

SS

Mizuno et al. 2006

Advection of Magnetic Field Loop• 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

No advection

Advection

No advection Advection

2D

3D

B2

B2

Volume-averaged magnetic energy density (2D)

Nx=512

256

128

1D Bondi Accretion1D Bondi flow with radial B-field (=1)=4/3, rc=8rg

ur

r/rg

r/rg r/rg

vr

Results follow initial Bondi flow structure

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

Resistive Relativistic MHD

Ideal / Resistive RMHD EqsIdeal 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: => infinity)

Charge current disappear in the Ohm’s law(degeneracy of equations, EM wave is decupled)

Numerical Integration Resistive RMHD

Hyperbolic equations

Source term

Stiff term

Constraint

Solve Relativistic Resistive MHD equations by taking care of 1. stiff equations appeared in Ampere’s law2. constraints ( no monopole, Gauss’s law)3. Courant conditions (the largest characteristic wave speed is always light speed.)

Difficulty of RRMHD1. Constraint

should be satisfied both

constraint numerically

2. Ampere’s law

Equation becomes stiff at high conductivity

ConstraintsApproaching Divergence cleaning method (Dedner et al. 2002, Komissarov 2007)

Introduce additional field & (for numerical noise)advect & decay in time

Stiff Equation

Problem comes from difference between dynamical time scale and diffusive time scale => analytical solution

Ampere’s lawdiffusion (stiff) term

Komissarov (2007)

Analytical solution

Operator splitting method

Hyperbolic + source termSolve by HLL method

source term (stiff part)Solve (ordinary differential) eqaution

Flow Chart for Calculation (RRMHD)

Step1: integrate diffusion term in half-time step

Step2: integrate advection term in half-time step

Step3: integrate advection term in full-time step

Step4: integrate diffusion term in full-time step

Un=(En+1/2, Bn)

(En+1, Bn+1)=Un+1

Strang Splitting Method

1D Shock-Tube Test (Brio & Wu)• Aim: Check the effect of resistivity (conductivity)

• Simple MHD version of Brio & Wu test

• (L, pL, ByL) = (1, 1, 0.5), (R, pR, By

R)=(0.125, 0.1, -0.5)

• Ideal EoS with =2Orange solid: =0Green dash-two-dotted: =10Red dash-dotted: =102

Purple dashed: =103

Blue dotted: =105

Black solid: exact solution in ideal RMHD

Smooth change from a wave-like solution (=0) to ideal-MHD solution (=105)

Relativistic Magnetic

ReconnectionAssumption• 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

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 -law, polytropic, and approximate Equation of

State (Synge-type)• Parallel computing (based on MPI)