  We formulate an effective numerical scheme that can readily, and accurately,
calculate the dynamics of weakly interacting multi-pulse solutions of the
quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The
scheme is based on a global centre-manifold reduction where one considers the
solution of the QCGLE as the composition of individual pulses plus a remainder
function, which is orthogonal to the adjoint eigenfunctions of the linearised
operator about a single pulse. This centre-manifold projection overcomes the
difficulties of other, more orthodox, numerical schemes, by yielding a
fast-slow system describing 'slow' ordinary differential equations for the
locations and phases of the individual pulses, and a 'fast' partial
differential equation for the remainder function. With small parameter
$\epsilon=e^{-\lambda_r d}$ where $\lambda_r$ is a constant and $d>0$ is the
pulse separation distance, we write the fast-slow system in terms of
first-order and second-order correction terms only, a formulation which is
solved more efficiently than the full system. This fast-slow system is
integrated numerically using adaptive time-stepping. Results are presented here
for two- and three-pulse interactions. For the two-pulse problem, cells of
periodic behaviour, separated by an infinite set of heteroclinic orbits, are
shown to 'split' under perturbation creating complex spiral behaviour. For the
case of three pulse interaction a range of dynamics, including chaotic pulse
interaction, are found. While results are presented for pulse interaction in
the QCGLE, the numerical scheme can also be applied to a wider class of
parabolic PDEs.
