  The second-order extended stability Factorized Runge-Kutta-Chebyshev (FRKC2)
class of explicit schemes for the integration of large systems of PDEs with
diffusive terms is presented. FRKC2 schemes are straightforward to implement
through ordered sequences of forward Euler steps with complex stepsizes, and
easily parallelised for large scale problems on distributed architectures.
  Preserving 7 digits for accuracy at 16 digit precision, the schemes are
theoretically capable of maintaining internal stability at acceleration factors
in excess of 6000 with respect to standard explicit Runge-Kutta methods. The
stability domains have approximately the same extents as those of RKC schemes,
and are a third longer than those of RKL2 schemes. Extension of FRKC methods to
fourth-order, by both complex splitting and Butcher composition techniques, is
discussed.
  A publicly available implementation of the FRKC2 class of schemes may be
obtained from maths.dit.ie/frkc
