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Making the Constraint Hypersurface and Attractor in Free Evolution

Making the Constraint Hypersurface and Attractor in Free Evolution. David R. Fiske Department of Physics University of Maryland. Advisor: Charles Misner. gr-qc/0304024. Overview. The Problem Evolution v. Constraints Free Evolution Method of Correction Adding Terms to Evolution Equations

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Making the Constraint Hypersurface and Attractor in Free Evolution

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  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 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 PSU Numerical Relativity Lunch

  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 PSU Numerical Relativity Lunch

  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!!! PSU Numerical Relativity Lunch

  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.) PSU Numerical Relativity Lunch

  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 PSU Numerical Relativity Lunch

  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)] PSU Numerical Relativity Lunch

  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. PSU Numerical Relativity Lunch

  9. Example: Simple Harmonic Oscillator PSU Numerical Relativity Lunch

  10. Example: Simple Harmonic Oscillator Correction Piece Underlying Formalism Piece PSU Numerical Relativity Lunch

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

  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) PSU Numerical Relativity Lunch

  13. “ADM” Maxwell PSU Numerical Relativity Lunch

  14. “BSSN” Maxwell “Grand Constraint” PSU Numerical Relativity Lunch

  15. “BSSN” Maxwell PSU Numerical Relativity Lunch

  16. Constraint Propagation • Evolution equations for the constraints: • Fourier Analysis: PSU Numerical Relativity Lunch

  17. Particular Solution Solutions for other wave numbers and other values of the parameters also show decay! PSU Numerical Relativity Lunch

  18. System I Primary Constraint PSU Numerical Relativity Lunch

  19. System II Primary Constraint PSU Numerical Relativity Lunch

  20. System II Secondary Constraint PSU Numerical Relativity Lunch

  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 PSU Numerical Relativity Lunch

  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 PSU Numerical Relativity Lunch

  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. PSU Numerical Relativity Lunch

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