  Both theoretical and numerical studies of the Kuramoto-Sivashinsky equation
have mostly considered periodic boundary conditions. In this setting, the
Fourier decomposition of the solution is central to theoretical ideas, such as
renormalization group arguments, as well as to numerical solution, allowing for
the construction of accurate and efficient time-steppers using standard
pseudo-spectral methods. In contrast, fixed boundary conditions induce boundary
layers and necessitate the use of non-uniform grids, usually generated by
orthogonal polynomials. On such bases, numerical differentiation is
ill-conditioned and can potentially lead to a catastrophic blow-up of round-off
error. In this paper, we use ideas recently explored by Viswanath (J. Comput.
Phys. 251(2013), pp. 414-431) to completely eliminate numerical differentiation
and linear solving from the time-stepping algorithm. We use the Green's
function-based method to investigate elements of the Kuramoto-Sivashinsky
dynamics over a range of five decades of the viscosity.
