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Making the Constraint Hypersurface and Attractor in Free Evolution

This presentation discusses the problem of evolution versus constraints in numerical relativity, the method of correction, and the addition of terms to evolution equations. It also explores the history of similar attempts and provides examples. The speaker concludes with worries and future directions.

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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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