Working with a problem in flow-induced vibration that could, under certain circumstances, be chaotic in the physical world (ie not just the mathematical model was chaotic) led me to question when we could know that our numerical methods were reliable.
This is not a trivial problem, and is not yet completely solved. However, I was able to
make some significant progress, and it is now understood that a posteriori
verification is at least possible in the case that the numerical method is reliable; that is,
after the computation has been done (or while it is being done), the reliability of the
method can be assessed, even in the case of a chaotic system (which is inherently
ill-conditioned by definition). The two key ideas are that of backward stability, or
exactly solving nearly the right problem, and ``well-enough conditioning'', which is
the proper extension of the notion of `well-conditioning' from numerical analysis. In
particular the idea of `well-posedness', more commonly known to mathematicians, is
now known not to be sufficient for a reliable computation. This notion is explained in
the paper `Defect Control for Chaotic Dynamical Systems' and more recently in the
pair of papers `Error Backward' and `What Good are Numerical Simulations of
Chaotic Dynamical Systems?'.