It is conceptually straightforward to extend the central idea of projective integration in time — which is to step only a few ``observables'' on the long time scale, rather than attempting to step forward all available quantities — to the spatial domain. Substantial progress has been made in this area by Gear and Kevrekidis, with particular emphasis on the conditions under which the resulting PDE solver is stable and accurate. Kevrekidis and Gear refer to the ``long space, short time'' scheme as ``gaptooth'' integration, and to the combination of space and time projective integration techniques ( ``long space, long time'') as ``patch dynamics''. It is instructive to illustrate how one might apply the patch dynamics algorithm to a fusion problem. We consider the problem of simulating a full tokamak cross section with kinetic electron and electromagnetic dynamics for a few energy confinement times.

Detailed gyrokinetic analyses of existing experimental data have shown that in ``interesting'' conditions (e.g., high β) one typically finds small scale features associated with electron and electromagnetic physics. As an example, a recent analysis of MAST profiles found unstable tearing modes at radial scales of approximately one ion gyroradius. It is not possible to resolve structures at this scale in a full-torus simulation. Nor it is possible to resolve Alfvén and electron transit time dynamics for a simulation lasting a few energy confinement times. Instead, we will consider calculating transport fluxes in a few annuli, with GS2 running on each space-time ``patch.''

GS2 simulations evolve distribution functions for each plasma species in three spatial dimensions, energy, magnetic moment, and time. For clarity, we illustrate the issues involved with a two-dimensional poloidal cross-section. Rather than attempting to simulate the entire cross-section, one selects a few flux surfaces. An annular region is cut out around each (in magnetic coordinates), reducing the size of the computational problem substantially.

For the transport problem, one can choose to simulate a flux-tube rather than the full annulus. This corresponds to taking a small region in the poloidal direction, and following the flux tube around for one or more full poloidal circuits of the field line. These ideas are currently fully implemented in GS2. The next step is to couple many flux-tubes together, projectively integrating macroscopic quantities forward in time. Such an approach is inherently highly parallelizable, and represents a unique opportunity to build fully kinetic transport modules that are accurate, stable and parallelizable, and thus suitable for full device simulation.