Constraint Preserving Boundary Treatment for the Einstein …jennseiler.com/docs/GR18.pdf · 2007. 10. 30. · Intro AEIHarmonic Harmonic SBP CP Results Conclusions Constraint Preserving

Intro AEIHarmonic Harmonic SBP CP Results Conclusions

Constraint Preserving Boundary Treatmentfor the Einstein Equations in 2nd Order Form

Jennifer Seiler

Collaborators:

B Szilagyi, L Rezzolla

Max-Planck-Institute for Gravitational PhysicsPotsdam, Germany

XXX Encuentros Relativistas EspanolesPuerto de la Cruz, Tenerife

19th September 2007

Jennifer Seiler [email protected] CP SBP Boundaries 2nd Order

Intro AEIHarmonic Harmonic SBP CP Results Conclusions IBVP

The Initial Boundary Value Problem

To simulate spacetimes numerically on a finite grid we truncate thecomputational domain by introducing an artificial outer boundary.

The boundary conditions should:

be compatible with the constraintsreduce reflectionsyield a well-posed initial-boundary value problem.



The AEIHarmonic Code

Generalized harmonic system

2nd differential order in space

Constraint damping

4th order finite differencing

Moving lego-excision

Mesh refinement (with Carpet)

Inspiral and Merger with Harmonic Coordinates. A

smooth crossing of the horizons can clearly be seen.


Intro AEIHarmonic Harmonic SBP CP Results Conclusions Description Features

”Generalised” Harmonic Coordinates

Coordinates:

GH coordinates, xµ, satisfy the condition �xµ = Γµ = Fµ.

Fµ(gαβ , xρ) as a source function chosen to fine tune gauge toaddress the requirements of specific simulations.

Provides solutions of the EEs provided that the constraints:

Cµ ≡ Γµ − Γµ =1√−g

∂

∂xκ

(√−ggλκ

)− Γµ = 0

and their time derivatives are initially satisfied.

Evolution Variables:

We define the evolution variables gµν ≡√−ggµν and

Qµν ≡ nρ∂ρgαβ , where nρ is timelike.

This simplifies the constraint equations to

Cµ ≡ − 1√−g

∂αgαµ − Γµ

and gives us a first order in time evolution system.


Intro AEIHarmonic Harmonic SBP CP Results Conclusions Description Features

Features of Generalized Harmonic Coordinates

System of equations is manifestly symmetric hyperbolic (givenreasonable metric conditions).

Simplifies the evolution equations:

When the gradient of this condition is substituted for terms inEinstein equations, the PP of each metric element reduces to asimple wave equation:

gγδgαβ,γδ + . . . = 0

Constraints have the same form.

The constraint equations may be incorporated into the generalizedharmonic coordinate conditions.

Gauge source terms for Harmonic coordinates allow free choice ofgauge for Einstein equations.



Summation by Parts Boundaries

The SBP method allows us to derive difference operators and boundary conditionwhich control the energy growth of the system and thus provide a mathematicallyand numerically well-posed system.A discrete difference operator is said to satisfy SBP for a scalar productE = 〈u, v〉 if the property

〈u,Dv〉+ 〈v ,Du〉 = (u · v) |ba

holds for all functions u, v in [a, b].One can construct a 3D SBP operator by applying the 1D operator to eachdirection. The resulting operator also satisfies SBP with respect to a diagonalscalar product

(u, v)Σ = hxhyhz

∑ijk

σijkuijk · vijk ,

Using SBP difference operators we can formulate an energy estimate for ourevolution system...





〈u,Dv〉+ 〈v ,Du〉 = (u · v) |ba


(u, v)Σ = hxhyhz

∑ijk

σijkuijk · vijk ,






〈u,Dv〉+ 〈v ,Du〉 = (u · v) |ba


(u, v)Σ = hxhyhz

∑ijk

σijkuijk · vijk ,






〈u,Dv〉+ 〈v ,Du〉 = (u · v) |ba


(u, v)Σ = hxhyhz

∑ijk

σijkuijk · vijk ,




Well-Posed Boundaries

For well-posedness, the energy estimate ξ(n) = ‖u (·, t) ‖2 of your system shouldsatisfy ‖u (·, t) ‖2 ≤ K (t) ‖u (·, 0) ‖2

We use the SBP rule to derive an estimate for the time derivative of the energy ofthe system.

Integrate using the SBP ruleSubstitute our boundary conditions and apply maximally dissipative condition.Applying that estimate as a penalty to our original evolution equationsWe can then choose coefficients for our boundary system which control the energygrowth of the whole system.

∂tQµν = − γ it

γttDi+Qµν − (γ ij +

γ itγjt

γtt)H−1(Aij + E0 − EN)Si )γ

µν

+2γ ij

γttβ0H−1E0i [(1 +

γ it

γtt)Di+γµν − Qµν

γtt+

2x

r2(γµν − g0)]

+2γ ij

γttβNH−1ENi [(1−

γ it

γtt)Di+γµν +

Qµν

γtt+

2x

r2(γµν − gN)]



Constraint Preservation

Sommerfeld-type outgoing conditions:(∂t − ∂x −

1

r

)(γµν − γµν

0 ) = 0

For CP Boundaries we set the four γtµ from the constraints:

Cµ = −∂tγtµ − ∂iγ

iµ − Fµ = 0

and we derive a set of outgoing conditions which specify the other 6metric components:(

∂x + ∂t +1

r

) (γAB − γAB

0

)= 0

(∂x + ∂t +

1

r

) (γtA − γxA − γtA

0 + γxA0

)= 0(

∂x + ∂t +1

r

) (γtt − 2γxt + γxx − γtt

0 + 2γxt0 − γxx

0

)= 0

see: [2] {Kreiss and Winicour, gr-qc 0602051}Jennifer Seiler [email protected] CP SBP Boundaries 2nd Order

Intro AEIHarmonic Harmonic SBP CP Results Conclusions High Shifts Robust Stability Tests Teukolsky Headon

Results for High Shifts

Scalar Waves log y

1e-10

1e-05

1

100000

1e+10

0 100 200 300 400 500

|g00

| ∞

t/M

SBP git=0.6Non-SBP git=0.6






Scalar Waves no log

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200

|g00

| ∞

t/M





Tests with Scalarwave testbed

Stability test for various shifts (0.6 <γ it

γtt< 1.1):

utt = −2γ it

γttuit −

γ ij

γttuij

Thin = Standard Somerfeld, Thick = SBP

Reflections for standard BCs clearly visible in right hand plot



Robust Stability Tests

Random Data + Brill

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0 200 400 600 800 1000 1200 1400 1600

||Cu || 2

t/M

BW with Rand CPSBP ||C0||2BW with Rand CPSBP ||C1||2BW CPSBP ||C0||2BW CPSBP ||C1||2

Random Data + Brill Wave

Random Kernel Amplitude = 0.1Brill Wave Amplitude = 0.5dx = 0.2 xmax = 7.1

Runs stable for in nonlinear regime for BrillWaves.

Stable for random data

Standard Sommerfeld type breaks rapidly forthis simulation

Checkerboard + Brill

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0 200 400 600 800 1000 1200

||Cu || 2

t/M

BW with Checkerboard CPSBP ||C0||2BW with Checkerboard Somm ||C0||2BW with Checkerboard SBP ||C0||2BW CPSBP ||C0||2

Checkerboard Data + Brill Wave

for each x(i), y(j), z(k) we add(−1)i+j+kA highest frequency noisepossibleChecker Kernel A = ±0.2Brill Wave Amplitude = 0.5dx = 0.2 xmax = 7.1

Standard sommerfeld seen in green (breaksquickly)



Results for Teukolsky/Brill Wave and Schwarschild Runs

Teukolsky

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0 100 200 300 400 500

||C0 || 2

t/M

Teuk CPSBP ||C0||2Teuk Somm ||C0||2Teuk SBP ||C0||2

High Amplitude Teukolsky WavesConstraint Norms for runs with:

Constraint Preserving ’SBP’ = Red

Pure SBP = Magenta

Standard sommerfeld-type = Blue

Schwarzschild

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

3.5e-05

4e-05

0 100 200 300 400 500 600 700

||Cu || 2

t/M

Schw CPSBP ||C0||2Schw CPSBP ||C1||2Schw Somm ||C0||2Schw Somm ||C1||2

Schwarzschild run with boundariestoo close in (40 M) forsommerfeld-type boundariesCPSBP remains stable



Head-on Runs with CPSBP

0

2e-07

4e-07

6e-07

8e-07

1e-06

1.2e-06

1.4e-06

0 20 40 60 80 100 120

||C0 || 2

t/M

HO Somm ||C0||2HO CPSBP ||C0||2

Headon Collison (mass 0.5, 2.5 Mseparation)

L2 Norm of Constraints for CPSBPvs regular boundaries

Significant improvement inconstraint preservation

1e-10

1e-08

1e-06

1e-04

0.01

1

100

0 50 100 150 200 250

xz/x

y cir

cum

fere

nce

ratio

t/M

HO CPSBP xz/xy-plane circumferencesHO Somm xz/xy-plane circumferences

Circumference ratios almostidentical

Some boundary effects are visiblefor the standard BC runs which arenot in the CPSBP run



Conclusions

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150 200 250

Q+ 20

t/M

HO CPSBP R=70 l=2 m=0HO Somm R=70 l=2 m=0

SBP provides a provably well-posed and demonstrably stable IBVPfor Generalized Harmonic evolutions on a Cartesian gridStands up to stability testsWe have developed a method which allows us to consistently useSBP on a Cartesian grid for corners and edges, and for a 2nd orderin space systemCPSBP provides a constraint preserving and noise reducingboundary system which is also demonstrable stable and well-posed



Thank You.


Excision Approaches Penalty Method Harmonic Constraints

[G. Calabrese and C. Gundlach, gr-qc 0509119]Discrete Boundary Treatment for the Shifted Wave Equation in Second OrderForm and Related Problems.General Relativity and Quantum Cosmology, 0509119, 31 July 2006.

[Kreiss and Winicour, gr-qc 0602051]Problems Which are Well-Posed in a Generalised Sense With Applications to theEinstein Equations.General Relativity and Quantum Cosmology, 0602051, 6 June 2006.

[G. Calabrese, J. Pullin, O. Reula, O. Sarbach, and M. Tiglio]Well Posed Constraint-preserving Boundary Conditions For the LinearizedEinstein Equations.Comm. Math. Phys., 240:377395, 5 Sept. 2002.

[H. Friedrich and G. Nagy, Comm. Math. Phys. 201]Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.Comm. Math. Phys., 201:619-655, 15 Sept. 1999.

[B. Szilagyi and J. Winicour, PRD 68:041501]Well-posed Initial-boundary Evolution in General Relativity.Phys. Rev. D, 68:041501(1)041501(5), 2003.


Excision Approaches Penalty Method Harmonic Constraints

Constraint Damping

The constraint equations are the generalized harmonic coordinateconditions: Cµ ≡ Γµ − Γµ = 0

constraint adjustment is done by the term

Aµν = CρAµνρ (xα, gαβ , ∂γgαβ)

in the evolution equations

∂α

(gαβ∂β gµν

)+ Sµν (g , ∂g) +

√−gAµν

+2√−g∇(µ F ν) − gµν∇αFα = 0.

Dissipation: f −→ f + ε(δijD+iD−i )w(δijD+iD−i )f where w is aweight factor that vanishes at the outer boundary. With D+iD−i

from blended SBP stencils.


Excision Approaches Penalty Method Harmonic

HarmonicExcision

(niD+i

)3f = 0 to all guard points, in layers stratified by length of the

outward normal pointing vector, from out to in.

LegoExcision with excision coefficientsxµ

rextrapolated around a smooth

virtual surface for the inner boundary.Radiation outer boundary conditions (i.e. outgoing only).



Past Work

[Stewart 1998] Necessary conditions for well-posedness of linearizedEinstein equations with constraint-preserving boundary conditions(Fourier-Laplace analysis)

[Friedrich & Nagy 1999] To-date the only formulation proven tosatisfy all the requirements for the fully nonlinear (vacuum) Einsteinequations (frame formalism)

[Kreiss & Winicour 2006] Well posed and constraint preservingboundary conditions for linearized Einstein Equations



Penalty Method

For the harmonic system the interior is:

∂tQµν =

γ it


γ itγjt

γtt)H−1Aijγ

µν

With the Boundaries it is:

∂tQµν = − γ it


γ itγjt

γtt)H−1(Aij + E0 − EN)Si )γ

µν

+2γ ij

γttβ0H−1E0i [(1 +

γ it

γtt)Di+γµν − Qµν

γtt+

2x

r2(γµν − g0)]

+2γ ij

γttβNH−1ENi [(1−

γ it

γtt)Di+γµν +

Qµν

γtt+

2x

r2(γµν − gN)]

Where γµν ≡√−ggµν and Qµν = g tα∂αγµν .


Excision Approaches Penalty Method Harmonic Description Evolution

”Generalised” Harmonic Coordinates

GH coordinates, xµ, satisfy the condition �xµ = Γµ = Fµ.

With the d’Alembertian, �φ ≡ 1√−g

∂

∂xλ

(√−ggλκ ∂φ

∂xκ

)GH coordinates coupled to the Einstein Equations gives:

Gµν = (Rµν −1

2gµνR) = 8πTµν =⇒

1

2gαβ∂α∂βgµν + gα(µ∂ν)Γ

α + Fµν(g , ∂g ) = 0

Gauge freedom from the ability to pick the four Γµ(gαβ , xρ).


Excision Approaches Penalty Method Harmonic Description Evolution

AEIHarmonic Evolution

We define the evolution variables gµν ≡√−ggµν and

Qµν ≡ nρ∂ρgαβ , where nρ is timelike.

This simplifies the constraint equations to

Cµ ≡ − 1√−g

∂αgαµ − Γµ

The AEIHarmonic code implements the first order in time system:

∂t gµν = − g it

g tt∂i g

µν +1

g ttQµν

∂tQµν = −∂i

((g ij − g itg jt

g tt

)∂j g

µν

)− ∂i

(g it

g ttQµν

)+ Sµν


Documents

Constraint Preserving Boundary Treatment for the Einstein …jennseiler.com/docs/GR18.pdf · 2007. 10. 30. · Intro AEIHarmonic Harmonic SBP CP Results Conclusions Constraint Preserving