making the constraint hypersurface and attractor in free evolution

May 2, 2003 PSU Numerical Relativity Lunch 1

Making the Constraint Hypersurface and Attractor in

Free Evolution

David R. Fiske

Department of Physics

University of Maryland

Advisor: Charles Misner

gr-qc/0304024

May 2, 2003 PSU Numerical Relativity Lunch

2

Overview

• The Problem• Evolution v. Constraints

• Free Evolution

• Method of Correction• Adding Terms to Evolution Equations

• History of similar attempts

• Examples (SHO and Maxwell)

• Conclusions, Worries, and Future Directions


3

Systems with Gauge Freedom Have Constraints

• Some of the PDEs tell how to make time updates

• Some of the PDEs constrain which initial data is allowed

• Analytically constraints are conserved• Numerically truncation violates constraints

BE

tE

B

t

0 E 0 B


4

Free Evolution

• Solve initial data problem

• Evolve via the evolution equations

• Monitor, but do not enforce, the constraints

THIS ALLOWS FORMALISM DEPENDENT, NON-PHYSICAL DYNAMICS TO INFLUENCE

STABILITY!!!


5

Changing Off-Constraint Behavior

• Can change off-constraint dynamics by adding terms to the evolution equations

• This does not change physics if f(0) = 0

• If f is chosen “wisely” this could improve the off-constraint dynamics. (Otherwise it could make them worse.)

)()()( CfqFqqFq tt


6

Some History

• Detweiler (1987)• Tried to fix the sign of the right hand side of the

constraint evolution equations

• Succeeded for special cases

• Brodbeck, Frittelli, Hübner, Reula (1999)• Embed Einstein equations into larger system

• For linear perturbations in constraints, it is mathematically stable


7

Some More History

• Yoneda and Shinkai (2001, 2002)• Add terms linear in constraints and derivatives

of constraints

• Perform eigenvalue analysis on principle parts

• Select terms with favorable eigenvalues

• Some terms successfully applied, others not [c.f. Yo, Baumgarte, Shapiro (2002)]


8

My Wish List for an Approach

• A constructive prescription for generating correction terms

• No dependence (if possible!) on perturbation theory

• Mathematically rigorous theory for believing the terms should work.


9

Example: Simple Harmonic Oscillator

x

v

v

x

dt

d

)0()()(),( 22 EtvtxvxC

v

Cx

x

Cv

v

C

dt

dv

x

C

dt

dx

dt

dC

22

222


10

Example: Simple Harmonic Oscillator

2

2

C

CK

x

v

v

x

dt

d

v

x

)0()()(),( 22 EtvtxvxC

222222

222

v

C

x

CK

v

Cx

x

Cv

v

C

dt

dv

x

C

dt

dx

dt

dC

UnderlyingFormalism

Piece

CorrectionPiece


11

Partial Differential Equations

• For PDEs, I need to take variational derivatives instead of partials

• I took the Maxwell Equations as a test case

2C

KFdt

dF

dt

d

xdCC N22


12

Formalisms of the Maxwell Equations

• As with the Einstein equations, there is more than one formalism of the Einstein equations

• Knapp, Walker, and Baumgarte (2002) investigated two Maxwell formulations similar to the “standard ADM” and BSSN formulations of Einstein (gr-qc/0201051)


13

“ADM” Maxwell

iiE

kkiikkit

iiit

EC

AAE

EA

EiEkkiikkit

iiit

CKAAE

EA

2


14

“BSSN” Maxwell

kkt

iikkit

iiit

AE

EA

iiE EC ii AC

222 wCCC E

“Grand Constraint”


15

“BSSN” Maxwell

kkt

iikkit

iiit

AE

EA

iiE EC ii AC

222 wCCC E

CKK

CKAE

CKKEA

Ckkt

EiEiikkit

iACiiit

2

2

2


16

Constraint Propagation

• Evolution equations for the constraints:

• Fourier Analysis:

CKCKKCC

CKCC

iACEt

EEiEt

2

2

2

2

CKkKKCC

CkKCkC

ACEt

EEEt

2

22

2

2


17

Particular Solution

xCxCxS

txSxCextC

txSxCextC

E

t

Et

E

,0,0

,0,

,0,3

3

1 kKKK CE

Solutions for other wave numbers and othervalues of the parameters also show decay!


18

System I Primary Constraint


19

System II Primary Constraint


20

System II Secondary Constraint


21

Conclusions

• Using the procedures presented here, different formulations of Maxwell’s equations were made to preserve the constraints asymptotically

• To the extent that the Maxwell-Einstein analogy holds, this is a positive sign for numerical relativity


22

Worries

• The correction terms change the order of the differential equations. Einstein (in ADM or BSSN form) will acquire fourth spatial derivatives!

• Linearized analysis (preliminary) of Einstein looks good, but nothing can be said for the full, non-linear equations


23

Future Directions

• Application to a first order formulation of the Einstein system (no fourth derivatives)

• Study of well-posedness of the corrected first order system

• Evaluation of some of the simpler terms generated for the BSSN system.

making the constraint hypersurface and attractor in free evolution

Documents

terms linear

constraint dynamics

constraint behaviorcan

constraint hypersurface

principle partsselect

evolution equationsmonitor

einstein equationsknapp

maxwell formulations