Trends in Numerical Relativistic Astrophysics

1
Trends in Numerical Relativistic Astrophysics

• David Meier
• Jet Propulsion Laboratory
• Caltech

2
Introductory Remarks

• Purpose of this lecture
• To discuss some of the numerical astrophysical
  research our group is doing now and in the near
  future
• To indicate where the field might go in the next
  10 years or so
• To stimulate thinking, especially in the younger
  members of the audience, on what types of
  problems and techniques may be addressable in the
  next 1/3 of a century
• Outline
• New astrophysics, same physics
• New physics, new astrophysics
• New numerical techniques on present-day computers
• New numerical techniques on future machines
• NOTE few equations will be presented refer to
  IPAM tutorial on relativistic astrophysics (Meier
  2005a)

3
I. New Astrophysics, Same Physics

• Numerical simulation of magnetized cosmic jets
• Non-relativistic flow
• Relativistic flow
• General Relativistic MHD of accretion disks with
  cooling
• Optically thin cooling magnetospheres jet
  production
• Optically thick cooling standard thin, cool
  disks
• Evolving GRMHD and neutron star mergers effects
  of magnetic fields

4
Cosmic Jets

• Occur in many types of objects

DYING DEAD STARS
CRAB PULSAR

• Often show wiggles

5
Non-Relativistic MHD Simulations of Cosmic Jets
in Decreasing Density Galactic Atmospheres
(Nakamura Meier 2004)

• Jet is more stable if density gradient is very
  steep ( ? z -3 ) and jet is mildly PFD (60)
• Jet is more unstable to m 1 helical kink if
  density gradient is shallow ( ? z -2 ) and jet is
  highly Poynting-flux dominated (90)
• (helical kink or m1 current-driven or
  screw instability)

6
Non-Relativistic MHD Simulations of Cosmic Jets
in Decreasing Density Galactic Atmospheres (cont.)

• Cause of the instability force-free magnetic
  field is unstable, but it can be stabilized if
  the plasma rotates (adds centrifugal force to
  radial force balance)

High magnetic field twist slow rotation ?
Unstable to helical kink
High magnetic field twist rapid rotation ?
More stable to helical kink
7
Relativistic MHD Simulations of Cosmic Jets in
Decreasing Density Galactic Atmospheres

• When flow is relativistic, new terms arise in the
  radial force equation, even without any plasma
  inertia (force-free!)
• When the magnetic field rotation rate v? ? c
• Conclusion Centrifugal force of the magnetic
  field inertia (?B) can balance pinch/hoop stress
• Question Are relativistic magnetized jets
  self-stabilizing, even without appreciable
  plasma? We will be investigating this shortly.

8
GRMHD Simulations of the MRI in Accretion Disks ?
WITH Optically Thin Radiative Cooling

• Present-day magneto-rotational instability disk
  simulations start with a thick torus with a weak
  magnetic poloidal field and differential
  (Keplerian) rotation
• (see IPAM 1 talks by Hawley, Gammie, Stone)
• Biggest deficiency of these simulations is the
  use of an adiabatic energy equation
  NO RADIATIVE COOLING
• Accreting material remains very hot (101113 K),
  but e radiate copiously above 1010 K
• IF ions couple well to e, the entire plasma will
  cool to 101011 K or less
• Predicted new structure will have a strong
  magnetosphere and jet, as seen in real objects

A turbulent dynamo develops in a few rotation
times (the MRI)
McKinney Gammie (2004)

• Question Does optically thin cooling produce
  a strong
• magnetosphere and jet as predicted by Meier
  (2005b)?
• Need to simulate MRI with GRMHD and cooling.

9
GRMHD Simulations of the MRI in Accretion Disks ?
WITH Optically Thick Radiative Cooling

• The first accretion disks to be modeled
  analytically (Shakura Sunyaev 1973) will be the
  last (and most difficult) ones to simulate
  numerically geometrically thin, optically thick
• Main scientific and computational issues
• Difficult radiative transfer
• Diffusion in disk interior
• Free streaming outside of disk
• Must handle photosphere carefully
• Coupling between radiation and plasma may be
  important in luminous disks
• Adaptive mesh refinement will be required to
  properly grid the disk

Geometrically Thin Torus
Geometrically Thick Torus

• Question Do simulated thin disks behave in
  the manner predicted by analytic
• theory? as observed in real systems?

10
EGRMHD Simulations of NS NS Mergers ? WITH
Magnetic Fields

• Virtually ALL observed neutron stars have
  magnetic fields (109 1015 G)
• Yet, NO present numerical relativity simulation
  of NS NS mergers includes magnetic fields in
  the dynamics
• Examples Rasio Shapiro (1999), Shibata et al.
  (2003), Baumgarte et al. (2004)
• Neutron stars merge, but do not form a black hole
• Instead, a differentially-rotating, super-NS forms

• How will the MRI change things (Balbus
  Hawley 1998)?
• MRI will create a strong viscosity, as in
  B.H. accretion disks
• Fastest growing MRI wavelength and growth
  rate (B Bi e?maxt)
• ?max ? VA ?Kep ?max 0.75 ?Kep
• Every rotation, magnetic field will increase
  by e?max?Kep 111
• Even the WEAKEST fields (109 G) will grow to
  dyn. important strengths (1016 G) in only 3-4
  rotation times of the VORTEX
• A differentially rotating, super-neutron-star
  likely will never form.
• It will collapse to a black hole in a few
  dynamical times.

• STRONG magnetic fields will likely lead to
  jets and ?-ray bursts
• Ignoring magnetic fields in relativistic
  astrophysical simulations is like simulating tar
  using alcohol. The understanding of B.H.
  formation critically depends in magnetic
  processes.

11
II. New Physics, New Astrophysics

• General relativistic charge dynamics
• Current sheets in pulsars
• Reconnection

12
General Relativistic Charge Dynamics

• GRCD equations (Meier 2004) were introduced in
  the IPAM tutorial on relativistic astrophysics,
  derived from the relativistic multi-fluid
  equations
• Fluid equations ?T ? ?mU 0
• ?T ? TFL TEMT 0
• TFL ? ?m(ep)/c2U?U U?H H?U/c2 p
  g
• TEM ? F ? F ¼ (F F) I / (4 ?)
• Charge equations ?T ? (?qU j)
  0
• ?T ? CT ?p2 (1 ? h?q)U hJ ? F/c ? ? J
  / (4 ?) (gen. Ohms law)
• C ?q(eqpq)/c2U?U U?j? j??U pq g
  (charge/beamed current tensor)
• GRCD will be important only on small scales (?x 1023 ?D), where 4 ?
  ?T ? CT/ ?p2 E
• For global simulations, ?x is 1057 times larger,
  so the standard Ohms law will suffice
• GRCD will be important primarily in simulations
  of small-scale phenomena or those with AMR.

13
Current Sheets and Reconnection Phenomena

• Pulsar Current Sheets
• Present simulations (Spitkovsky 2005) have
  finally solved the pulsar magnetosphere problem
  however, they do not resolve the current sheet
  (where B changes sign)
• The current sheet is believed to be a key to
  understanding the evolution and stability
  of the pulsar magnetosphere

• The GRCD equations have all the physics
  necessary to simulate the structure and
• evolution of the current sheet
• But, adaptive Mesh Refinement (AMR), of
  course, will be needed to resolve the current
  sheet
• One problem Plasma phenomena occur on very
  short time scales
• Solution solve the generalized Ohms law
  (?T ? CT ) as an implicit structure problem,
• setting ?/?t 0 in the Ohms law
  equations.
• With this technique, the current sheet will
  evolve on the pulsar time scale, not on the
  plasma
• time scale
• Supersonic Reconnection
• When the plasma shear flow is supersonic,
  ?U ?coll (particle collision frequency) and
  ?????? ??????the new Ohms Law terms will exceed
  the regular resistive term 4? ?T ? CT/ ?p2
  ? J
• This is another source of anomalous
  resistivity (scattering off shear flow layers)
• This, and other types of anomalous
  resistivity, may be important in reconnection

14
III. New Numerical Techniques for EGRD
(Numerical Relativity) On Present-day Machines

• Constrained Transport and Discrete Exterior
  Calculus
• Simple CT for MHD and full E M
• CT for EGRD (numerical relativity)
• Relation to DEC and lattice gauge simulations

15
Constrained Transport for MHD (Evans Hawley
1988)

• Magnetohydrodynamics (and full electrodynamics)
  is analogous to numerical relativity in that it
  has an evolution equation for the field and a
  constraint that must be satisfied
• B ?? c ? ? E ??B 0
• Modern MHD codes work by satisfying ??B O(?r) ?
  10-14 on the initial hypersurface and then using
  a differencing scheme that ensures
• ? (??B) / ?t ??B ??c ? ? ? ? E O(?r)
• How is this accomplished numerically?
• Answer by properly staggering the grid

?
?
B
??B ? 10-14
16
Constrained Transport for MHD (continued)

• The grid is staggered in space and time
• At t t0, B ? ? A and
• ??B Bx ? Bx ? By ? By ? Bz ? Bz?
• (Az4-Az2) (Ay4-Ay2) (Az3-Az1)
  (Ay3-Ay1)
• (Ax4-Ax2) (Az4-Az3) (Ax3-Ax1)
  Az2-Az1)
• (Ay4-Ay3) (Ax4-Ax3) (Ay2-Ay1)
  (Ax2-Ax1) ?? ? ? A O(?r)
• Then at t t½, similarly, ?? ? ? E O(?r) , so
  ??B O(?r)
• So, at t t1,
• ??B1 ??B0 ?t (??B½) O(?r)
• Proper staggering of the grid creates vector
  fields in which the divergence of a curl vanishes
  to machine accuracy, propagating the constraint
  (vector/Bianchi identity satisfied)

?
?
17
CT for Electrodynamics (Yee 1966 Meier 2003)

• Problem is more complicated in full
  electrodynamics
• Both of Maxwells equations and both of the
  constraints must be satisfied
• B ?? c ? ? E ??B 0
• E ? c ? ? B 4?J ??E 4??c
• creating the need for three interlaced updates
  (one each for B, E, and ?q)
• ?2? 4??q, E ?? B ? ? A
  Initialize
• E c ? ? B 4?J B ?? c ? ?
  E Evolve
• ?q ?? J
• ??E 4??q ??B 0
  ??E 4??q ??B 0 Implicit
• ?? E 4? ?? J ??B 0
  ?? E 4??q
• ?? E 4??q

?
?
?q
?
?
?
?
?
?
?
?
?
?
18
CT for Electrodynamics (continued)

• In covariant form, full electrodynamics appears
  much simpler
• F dA
  Initialize
• P ? (? ?F) 4? P ?J P ? (? ?F)
  0 Evolve
• ? ?J 0
• n ? (? ?F) 4? n ?J n ? (??F) 0 ? ?
  (? ?F) 0 Implicit
• 
  4-dimensional Bianchi identities
• Staggering the grid implicitly satisfies the
  Bianchi (and other simpler vector) identities to
  machine accuracy, and this implicitly propagates
  the constraints
• A staggered grid has deep geometric significance

J0
?
J2
19
Staggered Grids in 4-D

• The electrodynamics CT problem suggests a
  natural, simple, and elegant method for
  staggering finite difference grids in 4-D
• Scalars
• (hypercube
• corners)
• Vectors
• (hypercube edges)
• 2-Tensors
• (hypercube faces
• corners)

tn½
tn1
tn
?
?
?
?
?
?
J0
?
?
?
?
?
?
J3
?
?
?
?
J3
?
?
?
?
?
?
J2
J2
?
?
1
J0
J0
J1
J1
0
F12
?
?
?
?
?
?
F23
F13
F03
?
?
3
?
?
?
1
3
F02
?
F00, F11,
F01
0
20
Staggered Grids in 4-D (continued)
?121 ?222 ?323

• 3-Tensors,
• ?g, ?, etc.
• (hypercube edges
• cube body centers)
• 4-Tensors
• (hypercube corners,
• faces,
• body centers)

tn½
tn1
tn
?
?
?
?
?
?
?
?
?
?
?123
?
?
?
?
?
?
?012
?
?
?
?
?
?
?
?
?111 ?212 ?313
?
?
?
?
?
?
R0123
?
?
?
?
?
?
R0000 R1122 R2323 R3333
R0100 R1220
21
Staggered Grids in 4-D (continued)

• Some important features of generalized
  grid-staggering
• Special cases have interesting forms
• Kronecker delta (???) 4 ? 1s at cell corners
  12 ? 0s at cube faces
• Other identity tensors (?????, ??????? ) ?1 at
  corners 0 otherwise
• Levi-Civita tensor (?????) ? at
  hypercube body centers 0 otherwise
• Gives rise to the concept of a dual mesh and
  discrete differential forms
• Shift origin to hypercube-centered point to
  create the dual mesh
• ????? IS ????? as viewed from the dual mesh
• As viewed from the dual mesh the Maxwell tensor,
  the dual of F (M F), is simply F

tn½
tn1
tn
?
?
?
?
?
?
?0123
?
?
?
?
?
?
?
?
?
?
?
?
2
0
?
F13
?
?
M02
?
?
?
22
Staggered Grids in 4-D (continued)

• This is much more than just numerology
• Such techniques are strongly related to Discrete
  Exterior Calculus
  (the numerical method of the future)
• Outgrowth of Finite Element Method
• Currently under development in Engineering
  Mechanics
• Very formal mathematics (Differential Forms)
• Still in its early stages needs to deal with
• Vectors/Tensors
• Spacetime
• Curvature
• Also potentially related to Lattice Gauge
  Theories
• Quantum fields also are gauge fields with Bianchi
  identities
• Implementation on a staggered lattice satisfies
  these identities
• Differential forms correspond to
• Site variables (scalar, 0-forms)
• Link variables (vectors, 1-forms)
• Plaquette variables (2-tensors, 2-forms)
• All gauge field simulations have a common link
  
  using staggered grids to satisfy the
  Bianchi identities

Discrete Differential Forms
Dual Primal
Hirani 2003
Langfeld 2002
23
WARNING In Numerical Relativity CT is no
Substitute for a Stable Method

• Strongly vs. weakly hyperbolic scheme properties
  still must be respected
• Strongly hyperbolic schemes still promote
  stability
• Weakly hyperbolic schemes still promote
  instability
• CT is simply a very low-diffusive method of
  propagating gauge field constraints

Gauge wave evolved for 104 light-crossing times,
3 different resolutions, using strongly-hyperbolic
, spatially-differenced CT method (Miller Meier
2005)
Gauge wave evolved for 104 light-crossing times,
3 different resolutions, using weakly-hyperbolic,
spatially-differenced CT method (Miller Meier
2005)
24
IV. New Numerical Techniques On Future Machines

• Numerical Astrophysics in the past 70 years
• 70 years ago
• 35 years ago
• Today
• Numerical Astrophysics in the next 35 years

25
Numerical Astrophysics in the Past 70 Years

• 1935 (70 years ago)
• Stellar structure studied with polytropes and the
  Lane-Emden equation
• 1/r2 d/dr (r2 d? / dr) ?n p(r)
  p(0) ?n1 ?(r) ?(0) ?n
• Start at star center (r 0) and integrate until
  ? 0 (star surface) ? shooting method.
• Machine used adding machine (1 FLOP, 10
  BYTES).
• 1965 Gordon E. Moore of Intel Corp. writes
  famous article on exponential growth of computer
  chip densities (Moore 1965), to become known as
  Moores Law
• 1970 (35 years ago)
• Stellar structure studied with 2-point boundary
  value techniques shooting method only used to
  create initial guess for the solution, which
  was relaxed until entire structure converged
• Stellar collapse studied with state-of-the-art
  2-D MHD code, 40 x 40 grid, with adaptive
  shrinking mesh (LeBlanc Wilson 1970)
• Machines used CDC 6600/7600 (1 10 MFLOPs, 1
  MBYTE) ? 1067 times improvement
• 2003 Gordon E. Moore gives talk entitled No
  Exponential is Forever But We Can Delay
  Forever (Moore 2003), indicating that this
  pace can be sustained for at least another
  decade.
• 2005 (Today)
• Stellar collapse, disks, jets, explosions, etc.
  studied with state-of-the-art 3-D MHD codes, 300
  x 300 x 1000 grid
• Machines used 10 100 TFLOPs, 10 TBYTEs ?
  1067 times improvement
• Stellar structure, even rotation, still studied
  with the quasi-spherical approximation (1-D!!)

26
Numerical Astrophysics in the Past 70 Years

• Moore (1965) figures
• Moore (2003) figure

27
Numerical Astrophysics in the Next 35 Years

• 2040 (35 years from now) Three scenarios
• Current pace will top out at machines 100 times
  more powerful than now
• Current pace will continue unabated for 35 more
  years Machines 0.1 1 ZFLOPs, 0.1 1
  ZBYTE ? 1067 times improvement
• Quantum computing will come of age and cause a
  quantum jump in capability
• What could we do with a Zetta-FLOP/Zetta-BYTE
  machine?
• Suggested c.2040 astrophysics 1 more of the
  same
• Explicit, time-dependent simulations, more
  resolution, more time steps
• New problems allowed those where high
  resolution is needed throughout comp. domain
• However, 1067 times improvement only gives 100x
  better resolution in 3D, not enough to handle
  plasma phenomena, particle-particle interactions
• Also, AMR coupled with other higher-order
  techniques (e.g., spectral methods) will already
  have achieved very high accuracy in traditional
  explicit simulations
• More of the same is a naïve approach, and a
  poor use of the power of machines that will be
  available in future decades.

28
Numerical Astrophysics in the Next 35 Years

• Suggested c.2040 astrophysics 2a gridding (and
  adaptive gridding) in time
• Spacetime, after all, is a 4-D structure!
• Todays simulations will simply be tomorrows
  starting models for full 4-D structures
• The entire simulation, including all time steps
  and spacetime structure, will be kept in core
• Time grid will be refined for rapid evolution,
  just like spatial grid is for steep gradients
• Solution will be relaxed and converged for
• Accuracy in regions with high spatial and time
  derivatives
• Best gauge coordinate conditions at each point
  in the grid
• Best global gauge/coordinate conditions to avoid
  singularities
• Suggested c.2040 astrophysics 2b implicit
  multi-time-scale techniques
• Explicit schemes solve a marching
  problem qijkn1 qijkn fijk(q) ?t
• Implicit schemes relax F(q) over
  space-time F(q) ? ?q / ?t f(q) 0
• NOTES
• When ?t is small, time derivative of q is finite
  and solution evolves on short time scales
• When ?t is large, ?q/?t ? 0, F(q) ? f(q), and a
  static or steady-state structure is relaxed
• See Meier (1999) for
• suggested finite-element scheme using 4-D
  rectangular elements (should be updated using
  DEC)
• 4-D conservative, weak-form representations of
  all field and fluid equations, including Maxwell
  and Einstein fields
• Problems addressable

29
Talk Summary

• Short term (
• RMHD accretion disks, jet production, jet
  propagation
• EGRMHD effects of magnetic fields on B.H.
  formation and gravitational waves
• Mid-term (10 20 yr)
• GRCD current sheet structure and evolution,
  reconnection with generalized Ohms law
• CT/DEC for EGRD methods for numerical
  relativity more closely aligned with the
  mathematics behind the Einstein equations
  themselves (differential forms)
• Long-term (20 30 yr)
• Much faster computers
• MASDA (multidimensional astrophysical structural
  dynamical analysis) will be in use
• True 4-D spacetimes
• Multi-time-scale evolution
• Seamless multi-D stellar evolution from birth to
  black hole, GRB, and gravitational waves
• Note
• MASDA was the ancient chief Persian god (long
  before Islam)
• His prophet was Zarathustra (of 2001 A Space
  Odyssey fame)
• I expect MASDA to someday supplant ZEUS in the
  21st century, at least in numerical relativistic
  astrophysics

30
References

• Balbus, S.A. Hawley, J.F. 1998, Rev. Mod.
  Phys., 70, 1. Instability, turbulence, and
  enhanced transport in accretion disks.
• Baumgarte, T.W., Skoge, M.L., Shapiro, S.L.
  2004, Phys. Rev. D, 70, 064040. Black
  hole-neutron star binaries in general relativity
  Quasiequilibrium formulation.
• Evans, C.R. Hawley, J.F. 1988, Astrophys. J.,
  332, 659 677. Simulation of Magnetohydrodynamic
  Flows A Constrained Transport Method.
• Gammie, C.F. 2005, IPAM 1 Lectures, April 6.
  Relativistic Plasmas Near Black Holes General
  Relativistic MHD and Force-Free Electrodynamics.
  
• Hawley, J.F. 2005, IPAM 1 Lectures, April 6.
  General Relativistic Magnetohydrodynamic
  Simulations of Black Hole Accretion Disks.
• Hirani, A.N. 2003, Ph.D. thesis, Caltech,
  http//resolver.caltech.edu/CaltechETDetd-0520200
  3-095403. Discrete Exterior Calculus.
• Langfeld, K. 2002, Univ. Tubingen lecture, April
  2225, http//www.physik.unibas.ch/eurograd/Vorles
  ung/Langfeld/lattice.pdf. An Introduction to
  Lattice Gauge Theory.
• LeBlanc, J. Wilson, J.R. 1970, Astrophys. J.,
  161, 541 551. A Numerical Example of the
  Collapse of a Rotating Magnetized Star.
• McKinney, J.C. Gammie, C.F. 2004, Astrophys.
  J., 611, 977 995. A Measurement of the
  Electromagnetic Luminosity of a Kerr Black Hole.
• Meier, D.L. 1999, Astrophys. J., 518, 788 813.
  Multidimensional Astrophysical Structural and
  Dynamical Analysis. I. Development of a Nonlinear
  Finite Element Approach.
• Meier, D.L. 2003, Astrophys. J., 595, 980 991.
  Constrained Transport Algorithms for Numerical
  Relativity. I. Development of a
  Finite-Difference Scheme.
• Meier, D.L. 2004, Astrophys. J., 605, 340 349.
  Ohms Law in the Fast Lane General
  Relativistic Charge Dynamics.

31
References (cont.)

• Meier, D.L. 2005a, IPAM Tutorial Lectures, March
  11. Relativistic Astrophysics.
• Meier, D.L. 2005b, in From X-ray Binaries to
  Quasars Black Hole Accretion on All Mass Scales
  , proc. of the Amsterdam meeting, 1315 July
  2004, in press. Magnetically-dominated Accretion
  Flows (MDAFs) and Jet Production in the Low/Hard
  State.
• Miller, M.A. Meier 2005, in preparation.
  Constrained Transport Algorithms for Numerical
  Relativity. II. Hyperbolic Formulations.
• Moore, G.E. 1965, Electronics, 38, ?? ??.
  Cramming more components onto integrated
  circuits.
• Moore, G.E. 2003, talk at International Solid
  State Circuits Conference (ISSCC), 10 February
  2003. No Exponential is Forever but We Can
  Delay Forever .
• Nakamura, M. Meier, D.L. 2004, Astrophys. J.,
  617, 123 154. Poynting-Flux Dominated Jets in
  Decreasing-density Atmospheres. I. The
  Non-relativistic Current-driven Kink Instability
  and the Formation of Wiggled Structures.
• Rasio F.A. Shapiro, S.L. 1999, Class. Quant.
  Grav., 16, R1 R29. TOPICAL REVIEW
  coalescing binary neutron stars.
• Shakura, N.I. Sunyaev, R.A. 1973, Astron.
  Astroph., 24, 337 335. Black Holes in Binary
  Systems. Observational Appearance.
• Shibata, M., Taniguchi, K., Uryu, K. 2003,
  Phys. Rev. D, 68, 084020. Merger of binary
  neutron stars of unequal mass in full general
  relativity.
• Spitkovsky, A. 2005, in preparation.
• Stone, J.M. 2005, IPAM 1 Lectures, April 8.
  Studies of the MRI with a New Godunov Scheme for
  MHD.
• Yee, K.S. 1966, IEEE Trans. Antennas Propag.,
  AP-14, 302 307. Numerical solution of initial
  boundary value problems involving Maxwell's
  equations in isotropic media.