1. May 2, 2003PSU Numerical Relativity Lunch1 Making the Constraint Hypersurface and Attractor in Free Evolution David R. Fiske Department of Physics University of Maryland Advisor: Charles Misner gr-qc/0304024

2. May 2, 2003PSU Numerical Relativity Lunch2 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. May 2, 2003PSU Numerical Relativity Lunch3 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

4. May 2, 2003PSU Numerical Relativity Lunch4 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. May 2, 2003PSU Numerical Relativity Lunch5 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.)

6. May 2, 2003PSU Numerical Relativity Lunch6 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. May 2, 2003PSU Numerical Relativity Lunch7 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. May 2, 2003PSU Numerical Relativity Lunch8 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. May 2, 2003PSU Numerical Relativity Lunch9 Example: Simple Harmonic Oscillator

10. May 2, 2003PSU Numerical Relativity Lunch10 Example: Simple Harmonic Oscillator Underlying Formalism Piece Correction Piece

11. May 2, 2003PSU Numerical Relativity Lunch11 Partial Differential Equations For PDEs, I need to take variational derivatives instead of partials I took the Maxwell Equations as a test case

12. May 2, 2003PSU Numerical Relativity Lunch12 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)gr-qc/0201051

13. May 2, 2003PSU Numerical Relativity Lunch13 “ADM” Maxwell

14. May 2, 2003PSU Numerical Relativity Lunch14 “BSSN” Maxwell “Grand Constraint”

15. May 2, 2003PSU Numerical Relativity Lunch15 “BSSN” Maxwell

16. May 2, 2003PSU Numerical Relativity Lunch16 Constraint Propagation Evolution equations for the constraints: Fourier Analysis:

17. May 2, 2003PSU Numerical Relativity Lunch17 Particular Solution Solutions for other wave numbers and other values of the parameters also show decay!

18. May 2, 2003PSU Numerical Relativity Lunch18 System I Primary Constraint

19. May 2, 2003PSU Numerical Relativity Lunch19 System II Primary Constraint

20. May 2, 2003PSU Numerical Relativity Lunch20 System II Secondary Constraint

21. May 2, 2003PSU Numerical Relativity Lunch21 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. May 2, 2003PSU Numerical Relativity Lunch22 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. May 2, 2003PSU Numerical Relativity Lunch23 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.