What is PFASST?
PFASST is short for the Parallel Full Approximation Scheme in Space and Time and is an algorithm designed to provide parallelism in the time direction for the numerical solution of ordinary and partial differential equations. PFASST was first described in a paper by Emmett and Minion in 2012, which you can see here.
If you are familiar with the parareal algorithm for ODEs, then you could think of PFASST as a sort of parareal approach which differs from parareal in three important ways. First, instead of solving the ODE with coarse and fine propagators every iteration as in parareal, PFASST uses a deferred correction approach to improve the solution in time from the previous iteration. Second, the PFASST algorithm is set on an arbitrary hierarchy of levels on which the problem can be coarsened in space, time, order, or other ways as in the MLSDC algorithm. The coarse level problems are modified using a space-time FAS correction as in nonlinear multigrid methods. Third, the solution in time that PFASST converges to is equivalent to a fully implicit spectral collocation or Gauss Runge-Kutta method. Hence various implementations of PFASST do not require the user to supply a temporal integration method but rather only routines for explicit and implicit evaluation of the temporal derivative.
If you are familiar with multigrid methods, then PFASST could be thought of as a particular type of space-time multigrid method where the "smoother" in the time direction is a spectral deferred correction sweep. A paper by Bolten, Moser, and Speck describes PFASST from a multigrid perspective.
If you are familiar with spectral deferred correction methods, then PFASST is most easily thought of as a pipelined version of MLSDC with each time slice performing SDC sweeps in parallel.
In case you are wondering, the developers of PFASST pronounce PFASST with a silent "P" (like this major pharmaceutical company).