I've just finished Jörg Frauendiener's, "On the applicability of constrained symplectic integrators in general relativity". It's a readable paper (the necessary background is given in the paper) about numerical integration in Hamiltonian systems. The idea here is that if we have a symplectic manifold modelling the states of some system of ODE's, along with a hamiltonian we can give numerical integrators that respect the symplectic structure of the manifold. By this is meant that, for any point in the manifold and any time step, the numerical integrator gives a solution that approximates the state given by the hamiltonian flow up to a particular order. What exact it means that two points in a manifold can be approximated up to a sufficient order is not clear to me, but given that a point in the manifold is a function on some domain then I assume something like the

I've lately been interested in summation-by-parts (SBP) finite difference operators. These are matrices that replicate the integration by parts property of integration, when applied to functions over a grid. The SBP property ensures that numerically calculated solutions to DE's converge with time as well as with space. This is useful if one is interested in long-time evolutions. The SBP property ensures this by allowing discrete analogues of analytical techniques used when constructing energy estimates. The early papers justify SBP operators on the basis of the energy estimate that can be given. Whose not to say that there isn't a better class of finite difference operators out there, waiting for some one to stumble onto them? To start investigating this one needs some language in which all finite difference operators can be represented, classified and studied. Jörg's paper shows that symplectic manifolds can do this, at least for the very specific case he's interested in. Can it do it for other finite difference operators?

This question seems to be an obvious one (at least to me) and hence I expect there to be much literature about it...