# Stephen Grace and Xiangwei Tang

Λογισμικό & κατασκευή λογ/κού

13 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

99 εμφανίσεις

Stephen Grace and
Xiangwei

Tang

Background of Numerical Relativity

Numerical Relativity Equation

Interesting Characteristics

Parallel Computing Field

Solving Algorithms Used

Binary Black Hole Grand Challenge

Results

Current Issues

Why Black Hole

Black hole collisions should be strong generators

This is an extremely difficult problem for
numerical relativist.

To detect gravitational radiation and waves

Binary Black Holes (BBHs) are potentially a good
source of data

Europe and USA have mounted experiments to show
this

More topics than just BBHs

Binary Neutron Stars

Collapsing gravity fields, etc.

Uses Einstein’s General Relativity Equation as a
basis of study

There are hundreds of linear, nonlinear variations of
his original equation

G
uv

= 8πGT
uv

G
uv

Einstein Tensor (Space
-
time curvature)

T
uv

Stress
-
energy tensor (mass distribution)

G

Gravitational Constant

Can be quite small on the outside of the black hole, and quite large
on the interior

Indices
u,v

four index values

Corresponding to time and three spatial directions

Spatial directions are not specified in Polar, Cartesian, or Cylindrical
coordinates

This is the most basic form of the equation

All information from here on is with knowledge of the
nonlinear derivations

Total freedom in choosing the coordinate
system

Each coordinate system can provide different results
that are completely different from each other

Validity with different coordinate systems are
questionable

There are a variety of formulations of Einstein’s
equations

One potential formulation is a
constrained
Hamiltonian system (Hamiltonian Mechanics)

Forces are momentum invariant

12 equations: 6 spatial, 6
momenta

More sensitive sensors are requested to study
black holes and gravity

Modeling BBHs in parallel computing will help
refine sensors and look for anomalies

Simulating and handling the curvature
singularity inside the black hole

Interior of the black hole cannot affect the
exterior

Differing velocities of BBH collisions/mergers

Binary Black Hole Grand Challenge Alliance

Was started to solve Einstein’s Equation

Rules:

Develop problem solving for Nonlinear Einstein equations

Including dynamical adaptive multilevel parallel infrastructure

Provide controllable, convergent algorithms

The BBHGCA concluded with some major results

Actual simulation of a BBH collision

Organizations and groups still trying to solve the
problem based upon the initial findings of the
challenge

Algorithms used

Finite Difference

Most commonly found in potential solutions of the problem

Finite Element

Pseudo
-
spectral/spectral

Fourier Transforms used as an example

Can be highly accurate and use lower memory

Parallelization Techniques

-
Refinement

Multi
-
Grid

There are quite a few papers on each algorithm and
parallelization

Sadly, each paper differs so much that is hard to find a common
underlying equation to use as a basis

To solve the problem of the 3D spiraling
coalescence of two black holes

10 non
-
linear PDEs

4 Initial Value Equations (Elliptic)

6 Coupled Hyperbolic equations (Hyperbolic)

AMR is used to solve these equations.

Built by
Megware

for the
Albert Einstein Institute
(AEI) Numerical
Relativity Group (NRG)

Simulation video on next
slide was done using this
super computer

Ranked 192 on the
TOP500 list in 2007

Delivers efficiency of 80%

Frontends

2

Fileservers

17

Nodes

262

Primary Interconnect

Infiniband

Secondary Interconnect

Gigabit Ethernet

Storage Network

Gigabit Ethernet

Management Network

Fast Ethernet

Network Storage (Total)

140TB

CPUs (total) / Cores

524/1048 (Compute Nodes)

Processor Type

Intel Xeon 5160 Woodcrest
4MB Shared L2
-
Cache

RAM (total)

2096GB (Compute Nodes)

Video is of two black
holes of equal mass and
size colliding.

The various colors of red,
orange, and yellow are
gravitational waves
observed.

The white edges of the
box are to be the
boundaries of the
simulation

More development is proposed for gravitational sensors like the
LISA G.R.S.

Laser Interferometer Space Antenna with Gravitational Reference
Sensor

http://lisa.stanford.edu/LISAOverview.html

Scaling of the model

The simulation environment is not large enough to show all valid data

If the scaling is increased, supercomputers of today might not have enough
memory to run

Harmonic (Wave) evolution Scheme

Wave coordinates that satisfy the covariant scalar wave equations

Can simplify the equations greatly

More specific information:
http://arxiv.org/PS_cache/gr
-
qc/pdf/0512/0512093v3.pdf

G. Fox explains that by using the general
Einstein equation, tensors are used

High Performance Fortran is an attractive
language to use

Downside: need a Perl interface to adapt the
hierarchy to the language

Whenever the Fortran code is modified, the Perl
interface has to be re
-
written

Jack
Dongarra
, Ian Foster, Geoffrey Fox, William
Gropp

Ken
Kennedy Linda
Torczon
, Andy White. "Sourcebook of Parallel
Computing." San Francisco: Morgan Kaufmann Publishers,
2003. 195
-
199.

Supercomputers of the Max Planck Institute for Gravitational
Physics (
Damiana
)

http://supercomputers.aei.mpg.de/damiana/technical
-
specifications
-
1

http://supercomputers.aei.mpg.de/

AEI Numerical Relativity Group (BBHs Video)

http://numrel.aei.mpg.de/Visualisations/Archive/BinaryBla
ckHoles/GrazingBlackHoles/GrazingBH.html

The Hamiltonian.
OpenCourseWare
.
http://ocw.mit.edu/ans7870/18/18.013a/textb
ook/chapter16/section03.html

Binary Black Hole Grand Challenge.