So, from what I understand it seems that strong and symmetric hyperbolicity of equations is only defined for systems with first order derivatives. The significance is that strong or symmetric hyperbolic systems of differential equations are “well posed” that is, roughly speaking, that they have unique solutions that depend continuously on the initial data (i.e. a small change in the initial data produces a small change in the solution). Of cause this is very important if one is running some numerical simulation since it ensures that no problems in the simulation can be the result of a problem with the equations (although there can be issues with the representation of the equations on the computer).
There is a formulation of Einstein’s equations, called the Baumgarte-Shapiro-Shibata-Nakamura or BSSN system. It is a system involving first order time derivatives and second order spatial derivatives and therefore it is not immediately obvious that the system should behave well in numerical simulations. But it does, the question is why?
Sarbach, Calabrese, Pullin and Tiglio solved this by showing in “Hyperbolicity of the Baumgarte-Shapiro-Shibata-Nakamura system of Einstein evolution equations” that the BSSN is equivalent to a subfamily of the Kidder-Scheel-Teukolsky (KST) formulation for the Einstein equations. Since the KST systems have first order spatial and first order time derivatives this equivalence allows one to state that the BSSN system has, in some sense, the strongly and symmetrically hyperbolic properties as the KST system. Thus for certain values of certian variables the BSSN system is well posed, that is the BSSN system is a theoretically a good system to work with. The paper thus supports a variety of numerical evidence showing this (check out the paper for references).
All that being said I stumbled across Gundlach and Martin-Garcia’s, “Symmetric hyperbolic form of systems of second-order evolution equations subject to constraints” which attempts to give a definition of strong and symmetric hyperbolicity for such systems as the BSSN system. A quick search on ISI of the citing papers doesn’t bring anything up so it seems at first glance that their paper has gone no where at least with regards to the theory (Correct me if I’m wrong!).
In any case I find all this very interesting and am constantly surprised by the depth of complexity which means that no general theory for this kind of stuff has been worked out.