# A theoretical language to study numerical computation?

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 $L_2$ norm could be used. Such numerical integrators, called symplectic integrators, have some nice properties (see Jörg's paper for details). Apparently (when the paper was produced) there was a bit of a buzz about symplectic integrators that also respect constraints in the system. Jörg shows that the nature of the constraint system in GR is such that the current symplectic integration schemes won't work (he argues that they can never work). The problem is linked to the old issue that the hamiltonian system + constraints for GR is such that there is no hamiltonian flow on the constraint manifold. In non-mathsy language that means that there is no "time".

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...