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The following is an outline of a few of the important problems impeding work on numerical simulations of binary coalescence. We hope that the se will promote some useful discussion. There are certainly other important probl ems, but these are the problems we think should be tackled first and which could benefit from theoretical analysis.

- Gauge Modes
- Grid sucking, coordinate shocks, focusing, grid stretch ing...

- Constraint Violating Modes
- Solutions of the continuum evolution equations that are ill-behaved away from the constraint hypersurface

- Numerical Instabilities
- Solutions of the numerical approximatio n that are ill-behaved.

When our simulations blow up, what needs to be fixed? We have several example s that the continuum evolution equations are at fault and we would like to have a more general framework for understanding which systems of evolution equations have problems.

- Examples using seemingly reasonable sets of
evolution equat
ions:
- "ADM"
- Alcubierre's analysis of why "Conformal ADM" works but "ADM" doesn't.

- "Einstein-Ricci"
- Scheel's analysis of how the constraints can be used in 1D to make Einstein-Ricci stable.

- "ADM"
- How do we analyze/recognize these problems?
- We have some simplified, linear analyses.
- We really need some more powerful tools .

Points that should be kept in mind when studying the stability of evolution schemes.

- Well-posedness is not enough:
- It only guarantees that errors grows no faster than exp onentially.

- The principal part isn't the only source of problems.
- Perturbative analyses should be done about
backgrounds othe
r
than flat space or Schwarzschild coordinates.
- How does choice of background (slicing) affect stabilit y?

It seems clear that stability may depend strongly on the gauge conditions use d to evolve the system. Can we make any statements about the classes of slicings or shift conditions within which a given evolution scheme is stable.

- This really means we need to know what coordinates
to use
to keep out of trouble.
- What boundary conditions do we use on elliptic gauge conditions at excision boundaries?
- What gauge conditions drive us to time-independent solu tions at late time?
- How do we extract information from the solution to tell us how to place the coordinates?

While we have separated **constraint violating modes** and
**gauge modes
**it isn't clear that they can be analyzed independently.

**
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