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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
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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
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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
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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!!!
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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
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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
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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)]
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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.
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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
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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
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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
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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)
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“ADM” Maxwell
iiE
kkiikkit
iiit
EC
AAE
EA
EiEkkiikkit
iiit
CKAAE
EA
2
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“BSSN” Maxwell
kkt
iikkit
iiit
AE
EA
iiE EC ii AC
222 wCCC E
“Grand Constraint”
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“BSSN” Maxwell
kkt
iikkit
iiit
AE
EA
iiE EC ii AC
222 wCCC E
CKK
CKAE
CKKEA
Ckkt
EiEiikkit
iACiiit
2
2
2
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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
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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!
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System I Primary Constraint
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System II Primary Constraint
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System II Secondary Constraint
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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
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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
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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.