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Scripted foilset Detailed Discussion of Numerical Formulation and Solution of Collision 0f two Black Holes

Given by Geoffrey C. Fox at CPSP713 Case studies in Computational Science on Spring Semester 1996. Foils prepared 15 March 1996
Outside Index Summary of Material

This describes the structure of Numerical Relativity as a set of differential equations but it does discuss state of the art solvers involving adaptive meshes
Basic Motivation of General Relativity and its experimental tests
Metric Tensor, its derivatives and Einstein's equations
Initial value formulation and structure of elliptic and hyperbolic equations
Examination of particular finite difference scheme for the Wave equation in three dimensions -- a study to understand large distances issues in solving numerical relativity

Table of Contents for full HTML of Detailed Discussion of Numerical Formulation and Solution of Collision 0f two Black Holes

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1 CPS713 Module on Numerical Simulation of the Collision of two Black Holes as part of Case Study (II) on CFD and Numerical Relativity
2 Abstract of Module on Numerical Simulation of the Collision of two Black Holes
3 References for CPS713 Module on Numerical Simulation of the Collision of two Black Holes
4 The Spirit of General Relativity as a Description of Gravitational Forces as the Structure of Space-Time
5 General Relativity as a Theory of Distorted Space-Time
6 The Space-Time Structure Created by a Heavy Bowling Ball
7 The Path of a Marble in a Distorted Space-Time
8 Basic Notation for Numerical Formulation of Einstein's Equations
9 The Metric Tensor in Einstein's Formulation of General Relativity-I
10 The Metric Tensor in Einstein's Formulation of General Relativity-II
11 Why Study General Relativity Numerically
12 An Example of Gravitational Waveforms
13 Two Polarizations of Gravitational Waveforms
14 A schematic view of a LIGO Interferometer
15 Schematic Layout of the Initial LIGO facilities
16 Expected Total Noise in each of LIGO's first 4km interferometers
17 Expected Signal versus Noise in Gravitational Wave Detectors
18 Some Tests of General Relativity
19 More Tests of General Relativity
20 Equivalence Principle
21 Initial Value Formulation of General Relativity
22 Projection of Einstein's Equations onto Spacial Surfaces
23 Structure of Einstein's Equations in Initial Formulation
24 Linearization of Time Evolution Equations for q ij
25 Structure of Numerical Relativity Equations in terms of 3 by 3 matrices q and K
26 Coodinate and Foliation Choices in General Relativity
27 The Lapse and Shift in Gauge Transformations
28 Geometrical Picture for Lapse and Shift Gauge Transformations
29 Notation for Einstein's Equations in Initial Value Formulation
30 The Four Elliptic Constraint Equations in Initial Value Formulation of Einstein's Equations
31 The Twelve Hyperbolic Evolution Equations in Initial Value Formulation of Einstein's Equations
32 A benchmark Numerical Relativity problem
33 Characteristic Surfaces and Key Features of Pittsburgh Approach
34 Numerical Formulation of Three Dimensional Wave Equation in Polar Coordinates
35 Compactification and Computational Variables for Three Dimensional Wave Equation
36 Final Computational Formulation of Pittsburgh Benchmark
37 Final Computational Formulation of Pittsburgh Benchmark -- Diagram
38 Discretization of Computational Formulation of 3D Wave Equation
39 Finite Volume Integral Formulation of Differencing Equations
40 Stable Finite Difference Form of Discretized Pittsburgh Wave Equations-I
41 Stable Finite Difference Form of Discretized Pittsburgh WaveEquations-II

Outside Index Summary of Material

HTML version of Scripted Foils prepared 15 March 1996

Foil 1 CPS713 Module on Numerical Simulation of the Collision of two Black Holes as part of Case Study (II) on CFD and Numerical Relativity

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Geoffrey Fox
NPAC
111 College Place
Syracuse NY 13244-4100
HTML version of Scripted Foils prepared 15 March 1996

Foil 2 Abstract of Module on Numerical Simulation of the Collision of two Black Holes

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
This describes the structure of Numerical Relativity as a set of differential equations but it does discuss state of the art solvers involving adaptive meshes
Basic Motivation of General Relativity and its experimental tests
Metric Tensor, its derivatives and Einstein's equations
Initial value formulation and structure of elliptic and hyperbolic equations
Examination of particular finite difference scheme for the Wave equation in three dimensions -- a study to understand large distances issues in solving numerical relativity
HTML version of Scripted Foils prepared 15 March 1996

Foil 3 References for CPS713 Module on Numerical Simulation of the Collision of two Black Holes

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Introductory References:
  • "Solving Einstein's Equations on Supercomputers", Ed Seidel, NCSA,Illinois
  • "Numerical Relativity", Ed Seidel(NCSA), Wai-Mo Suen(Washington University, St. Louis)
More Technical Background for Discussion In Course:
  • "A Description of the Initial Value Formulation of Vacuum General Relativity for the Non-Specialist", Mark Miller(NPAC,Syracuse University)
  • "Gravitational Wave Extraction - A Benchmark?", R. Gomez, J. Winicour(Pittsburgh Univ.)
  • "Pittsburgh Group Code : Gravitational Wave Extraction - A Benchmark? - A Summary", Mark Miller(NPAC)
HTML version of Scripted Foils prepared 15 March 1996

Foil 4 The Spirit of General Relativity as a Description of Gravitational Forces as the Structure of Space-Time

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We are familiar with Newton's Laws for the Interactions between Particles:
"What" Causes this force?
  • At the most fundamental level of Quantum gravity, we will we find "elementary" Gauge Particles -- The Gravitons which are analogue of Photons for Electromagnetism and Gluons for Quantum Chromodynamics
  • Most Physicists believe all forces can be described by a unified theory at the Quantum level but this is still speculative.
  • The Graviton is unusual as has Spin 2 (Corresponding to two indices in metric tensor gmn we will soon study) wheras other force quanta are spin 1
    • For instance, spin 1 Photon corresponds to single index vector field Am
  • However we will study the classical theory which is sufficient for all observations
HTML version of Scripted Foils prepared 15 March 1996

Foil 5 General Relativity as a Theory of Distorted Space-Time

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Note that gravitational forces cause particles to move in vacuum but we are also familiar with particles moving "downwards" if placed on a nonflat surface
Einstein's theory of General Relativity is a brilliant generalization of this to describe all gravity in terms of structure of Space-Time
We do not say say the Earth creates a "Gravitational Force" which causes Apples to fall as noted by Newton
Rather The Earth distorts space-Time and Apples move in this distorted Space-Time
In Newton's law, gravitational force of Earth proportional to its Mass
In Einstein's description, distortion of Space-Time is proportional to Mass of particle
Everything (including you) distorts Space-Time;
  • This distortion is insignificant except for very Heavy Bodies
HTML version of Scripted Foils prepared 15 March 1996

Foil 6 The Space-Time Structure Created by a Heavy Bowling Ball

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We can consider the distortion of Space-Time due to a Heavy Body as analogous to distortion of a FLAT rubber sheet into a DROOPING rubber sheet when a bowling ball is placed in the Middle
HTML version of Scripted Foils prepared 15 March 1996

Foil 7 The Path of a Marble in a Distorted Space-Time

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Now place a much lighter body (a marble) near the bowling ball
  • The marble will roll to the "bottom" of the distortion in Space-Time
HTML version of Scripted Foils prepared 15 March 1996

Foil 8 Basic Notation for Numerical Formulation of Einstein's Equations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Greek indices m,n .. denote Space-Time and run from 0 (time) to 3.
Latin indices i,j,k .. denote Space and run from 1 to 3 (x y and z).
Note indices ALWAYS balance in equations and there is difference between upper and lower indices
  • For Instance matrix M ij is inverse of matrix M ij.
Again M ij M jk = d i k is Expression of Inverse Condition
HTML version of Scripted Foils prepared 15 March 1996

Foil 9 The Metric Tensor in Einstein's Formulation of General Relativity-I

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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The speed of light c is set to 1.
  • Normally 3 X 1010 cm/second
All quantities in General Relativity are a function of the four vector:
The dynamical variables of the theory are the 10 independent components of the the metric tensor:
For example, this describes distortion of rubber sheet by bowling ball
g mn = g nm is SYMMETRIC
If two Space-Time points are separated by four vector d m
Square of distance between them is g mn d m d n
HTML version of Scripted Foils prepared 15 March 1996

Foil 10 The Metric Tensor in Einstein's Formulation of General Relativity-II

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Each of 16 (10 independent) components of 4 by 4 tensor g mn is a function
of the four vector
Einstein's equations are nonlinear equations for the metric g and its derivatives usually constructed in terms of so called Christoffel Symbol G
HTML version of Scripted Foils prepared 15 March 1996

Foil 11 Why Study General Relativity Numerically

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Newtonian gravity is linear (weak field) limit of General Relativity
Tests of General Relativity --
  • Small deviations from Newton's laws in planetary orbits
  • Energy loss in Binary Neutron Stars from Gravitational wave radiation
  • Bending of light in strong gravitational fields
New experiments will directly observe gravitational waves due to interactions with sensitive detectors on earth
  • LIGO in Louisiana and Washington State(Hanford)
  • VIRGO in Europe
Can estimate that Computations of Gravitational Waves from black hole collision would take 100,000 hours on a Cray-YMP
  • 10 Hours on a Teraflop Machine
Estimate that LIGO will sensitivity to detect black hole gravitational waves around the year 2005
HTML version of Scripted Foils prepared 15 March 1996

Foil 12 An Example of Gravitational Waveforms

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Science 256,281-412, 17 April 92
Figure 1 shows two components of a gravitational wave versus time
  • An example of gravitational waveforms and the information they carry.
  • Each gravitational wave has two waveforms, dimensionless functions of time called h+(t) and hx(t).
  • The specific waveforms shown here are from the last few minutes or seconds of the spiraling together of a compact binary system (one made of two black holes, two neutron stars, or a black hole and a neutron star).
  • By monitoring these waveforms, LIGO can allow researchers to determine the binary's distance from Earth r, the masses of its two bodies or, equivalently, their total mass M and reduced mass m, and their orbital eccentricity e, and orbital inclination to the line of sight i.
  • To allow the determination of the eccentricity e, LIGO will measure the shapes of the individual waveform oscillations; note the shape shown on the upper right.
  • For the determination of i (when e = 0 for pedagogic simplicity), LIGO will measure the ratio of the amplitudes, h+ and hx see the formula in the lower right.
  • The parameters r, m and M determine (i) the waveforms' absolute amplitudes as they sweep past a frequency f:
  • hamp proportional to mu M^2/3 r^-1 f^2/3;
  • and (ii) the number of cycles n=f^2 (df/dt)^-1 that the waveforms spend near frequency f:
  • n**alpha (mu M^2/3 f^5/3)^-1 .
  • From hamp and n, LIGO can be used to determine r and mu M^2/3.
  • From mu M^2/3, and from late-time post-Newtonian facets of the waveform (not shown here)
  • or the frequency at which the inspiral terminates or both,
  • LIGO can be used to deduce the individual value of mu and M.
  • The simple inspiral waves shown here are modified at late times by post-Newtonian
  • and then fully relativistic effects and then are followed by much more complicated waveforms from the final collision or tidal disruption of the black holes or neutron stars.
  • It is these final relativistic, collision, and disruption waveforms that will bring LIGO the most interesting information.
HTML version of Scripted Foils prepared 15 March 1996

Foil 13 Two Polarizations of Gravitational Waveforms

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Science 256,281-412, 17 April 92
Figure 2 shows two components of a gravitational wave versus time
  • A general-relativistic gravitational wave propagating in a z direction squeezes and stretches the separation of test particles in a plane perpendicular to the z axis.
  • The wave acts by a combination of its two polarizations:
  • + ("plus") polarization pushes test particles together along the x direction and pushes them apart along the y direction when h+(t) is negative;
  • x ("cross") polarization pushes and pulls test particles,
  • as determined by the sign of hx (t) at 45 degree angle from the x and y axes.
HTML version of Scripted Foils prepared 15 March 1996

Foil 14 A schematic view of a LIGO Interferometer

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Science 256,281-412, 17 April 92
HTML version of Scripted Foils prepared 15 March 1996

Foil 15 Schematic Layout of the Initial LIGO facilities

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Science 256,281-412, 17 April 92
  • For each interferometer, the laser beam is conditioned in an input optics chain before it reaches the beam splitter.
  • After passing through the beam splitter, the beams enter the chambers containing the test masses, where they are directed into 2-km or 4-km interferometer cavity.
  • After leaving the cavity, the beams, now back in the test mass chamber, are directed back to the beam splitter and then into an output optics chain that terminates with the photodetector.
  • At site 1, all elements of the 4-km interferometer lie along the two arms,
  • whereas the beam splitter and input and output optics chains for the 2-km interferometer lie between the two arms.
HTML version of Scripted Foils prepared 15 March 1996

Foil 16 Expected Total Noise in each of LIGO's first 4km interferometers

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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(Upper solid curve) and in a more advanced interferometer (lower solid curve)
The dashed curves show various contributions to the first interferometer's noise.
Science 256,281-412, 17 April 92
HTML version of Scripted Foils prepared 15 March 1996

Foil 17 Expected Signal versus Noise in Gravitational Wave Detectors

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Comparison between rms noise h/rms in LIGO's first and advanced detector systems and the characteristic amplitude h/c of gravitational wave bursts from several hypothesized sources
  • NS is a Neutron Star
  • BH is a Black Hole
Science 256,281-412, 17 April 92
HTML version of Scripted Foils prepared 15 March 1996

Foil 18 Some Tests of General Relativity

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Einstein first published equations in 1915 with three predictions:
Bending of Light as it passed massive objects such as stars
  • Experimentally verified in 1919 by observation of star field near the sun during an eclipse
There is a small (43 seconds of arc per century -- with 3600 seconds of arc as one degree) shift in Mercury's perihelion not accounted for by Newtonian Gravity
  • Einstein verified that General relativity predicted this soon after publication of equations
"Blue" shift (frequency or energy change) of light as it falls down a gravitational field
  • In 1960, (5 years after Einstein's death) Pound and Rebka observed the two parts in 1012 frequency shift as photons fell 74 feet of the Jefferson tower of the physics building at Harvard
  • They used Mossbauer effect
HTML version of Scripted Foils prepared 15 March 1996

Foil 19 More Tests of General Relativity

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Observed time delay of radar signals bounced off planets or spacecraft during superior conjunction
  • so that signals are affected by close passage to sun
Observed decrease in period of binary pulsar discovered by Hulse and Taylor(1975)
  • Consistent with energy decrease due to radiation of gravitational waves predicted by general relativity
A laser monitors distance between Moon and Earth showing that they both have same acceleration due to sun with a precision of 7 parts per ten trillion(1010)
  • Tests Equivalence Principle
HTML version of Scripted Foils prepared 15 March 1996

Foil 20 Equivalence Principle

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Says that a uniform gravitational field is the same as constant acceleration
  • If you are in a room with no windows, there is no way for you to tell that you were on earth and subject to gravitational field as opposed to hypothesis that you were far from any field but accelerating at 9.8 meters/sec2
Alternatively, Equivalence Principle says that the gravitational mass (the mass m1 that appears in Newton's law of gravity)
  • F=Gm1m2/r122
is equal to inertial mass m1 appearing in Newton's law of motion
  • F=m1 x acceleration
In general relativity, the Equivalence Principle is built in
HTML version of Scripted Foils prepared 15 March 1996

Foil 21 Initial Value Formulation of General Relativity

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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We formulate General Relativity in fashion one would expect for solving wave equations in three spatial dimensions
  • This is called foliation and slices Space-Time into
  • Spacial Surfaces labelled by particular time values t and coordinatized by (x,y,z)
HTML version of Scripted Foils prepared 15 March 1996

Foil 22 Projection of Einstein's Equations onto Spacial Surfaces

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
The Einstein equations for g mn can now be written in terms of a reduced 3 by 3 symmetric metric matrix q ij
q ij has 9 components of which 6 are independent
HTML version of Scripted Foils prepared 15 March 1996

Foil 23 Structure of Einstein's Equations in Initial Formulation

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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There are 10 dynamical equations for 6 independent components of q ij
  • 4 of these are elliptic constraint equations which must be satisfied by initial data -- analogous to initial conditions on positions and velocities in Newton's equations.
  • 6 of the equations are true dynamical equations describing time evolution of the q ij
HTML version of Scripted Foils prepared 15 March 1996

Foil 24 Linearization of Time Evolution Equations for q ij

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
When Solving a second order time evolution equation such as
One typically converts this to a pair of first order in time equations by transformation:
We perform the analogous step in general relativity, introducing a new symmetric matrix K ij which is called the extrinsic curvature and related to but not exactly equal to single time derivative of q ij
HTML version of Scripted Foils prepared 15 March 1996

Foil 25 Structure of Numerical Relativity Equations in terms of 3 by 3 matrices q and K

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
K is a Symmetric 3 by 3 matrix
And we now have 14 equations:
HTML version of Scripted Foils prepared 15 March 1996

Foil 26 Coodinate and Foliation Choices in General Relativity

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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One of the fascinating features of Numerical Relativity is the freedom to choose coordinates
  • This can allow equations to be smooth and appropriate for finite differences in one choice and very irregular and requiring adaptive finite element in another choice
  • Choice is called Choice of Gauge:
HTML version of Scripted Foils prepared 15 March 1996

Foil 27 The Lapse and Shift in Gauge Transformations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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The Lapse describes how much time elapses between two spacial surfaces
The Shift functions N i describe the tangential shift as you move from the t 0 to t 0+ dt spacial surface
Choose Lapse and Shift to:
  • Avoid Unphysical Singularities (One cannot avoid physical black hole singularity
    • Choice of spherical polar coordinates has singularity at radius r=0.
  • Keep Numerical Algorithm Stable
  • Insure Consistency of Constraints
HTML version of Scripted Foils prepared 15 March 1996

Foil 28 Geometrical Picture for Lapse and Shift Gauge Transformations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
a and N i are depicted below for two adjacent spacial surfaces t and t + dt
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Foil 29 Notation for Einstein's Equations in Initial Value Formulation

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Define Inverse q ij of 3 by 3 matrix q ij:
Define the spacial restriction of Christoffel Symbol by:
  • Note summation convention over m in above formula.
Get full Christoffel Symbol by replacing q ij by g mn and makes all indices Greek running from 0 to 3.
HTML version of Scripted Foils prepared 15 March 1996

Foil 30 The Four Elliptic Constraint Equations in Initial Value Formulation of Einstein's Equations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Note these are very nonlinear and complicated
  • Written out as Fortran code, they will be very complicated
  • Need high level (symbolic Algebra) front end
HTML version of Scripted Foils prepared 15 March 1996

Foil 31 The Twelve Hyperbolic Evolution Equations in Initial Value Formulation of Einstein's Equations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
6 First Order in Time Evolution Equations for q ij
6 First Order in Time Evolution Equations for K ij
HTML version of Scripted Foils prepared 15 March 1996

Foil 32 A benchmark Numerical Relativity problem

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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The numerical relativity problem has several key features
Our first example is based on work of University of Pittsburgh group involving R. Gomez and J.Winicour
  • R. Gomez, R. Isaacson and J. Winicour, Journal Computational Physics, 98,11(1992)
  • R. Gomez, J. Winicour, "Gravitational Wave Extraction - A Benchmark ?"
This studies the wave-like features of Einstein's equations where key numerical problem is reliable extraction of Gravitational waves
This is nontrivial as gravitational waves can only be extracted after one has evolved a "long way" from the black holes
It is difficult numerically to reliable evolve oscillating solutions through long distances
  • See examples of convection equations in Hirsch
HTML version of Scripted Foils prepared 15 March 1996

Foil 33 Characteristic Surfaces and Key Features of Pittsburgh Approach

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Key features of Pittsburgh Approach
  • Use Characteristic Surfaces instead of spacial surfaces
    • Spacial: fixed time
    • Characteristic: fixed t+r or t-r
  • Compactification -- Map infinity (where waves sought) to finite point)
HTML version of Scripted Foils prepared 15 March 1996

Foil 34 Numerical Formulation of Three Dimensional Wave Equation in Polar Coordinates

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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The 3D Wave
Equation is:
We will study this as "essential issues" in wave extraction are exhibited
  • More generally take position and partial derivative dependent coefficients of second derivatives and add such forcing terms
Transform to Spherical
Polar Coordinates
Introduce characteristic
    • variables:
HTML version of Scripted Foils prepared 15 March 1996

Foil 35 Compactification and Computational Variables for Three Dimensional Wave Equation

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
Introduce new computational variables (x,h,z)
  • Note x=1 corresponds to radial variable r equal to infinity.
These three variables are used as well as u (=t-r) replacing time as fourth independent variable
Replace F by G as dependent variable:
HTML version of Scripted Foils prepared 15 March 1996

Foil 36 Final Computational Formulation of Pittsburgh Benchmark

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We must solve:
In Compact Domain:
With Boundary Conditions given in diagram:
Note we assume that you have used other techniques (represntations) to integrate upto:
HTML version of Scripted Foils prepared 15 March 1996

Foil 37 Final Computational Formulation of Pittsburgh Benchmark -- Diagram

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
This illustrates formalism on previous page
HTML version of Scripted Foils prepared 15 March 1996

Foil 38 Discretization of Computational Formulation of 3D Wave Equation

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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We take simple equally paced gridpoints in u, x, h, and z.
We have already seen for convection equation subtly in time discretization of this type of equation.
A stable explicit method can be obtained by a carefully chosen integral formulation described in Miller's memo.
This involves integrating over a box PQRS with sides parallel to the characteristic axes u=constant, v=constant
HTML version of Scripted Foils prepared 15 March 1996

Foil 39 Finite Volume Integral Formulation of Differencing Equations

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
Full HTML Index
We find these pictures for box PQRS in (t,r) (u,v) (u,x) spaces
HTML version of Scripted Foils prepared 15 March 1996

Foil 40 Stable Finite Difference Form of Discretized Pittsburgh Wave Equations-I

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Alternatively we can view as a brilliant choice of finite difference scheme
Which is justified a posteriori by the Von Neumann stability analysis given in Pittsburgh memo which shows all eigenvalues have unit modulus as required for
  • Stable algorithm preserving magnitude of amplitude.
This clever choice is:
Giving:
HTML version of Scripted Foils prepared 15 March 1996

Foil 41 Stable Finite Difference Form of Discretized Pittsburgh WaveEquations-II

From Numerical Formulation and Solution of Collision 0f two Black Holes CPSP713 Case studies in Computational Science -- Spring Semester 1996. *
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Further we similarily:
Which gives final numerical equations:
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