Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation
T. MacKenzie and A. J. Roberts
(Received 7 August 2000)
Abstract
We analyse the nonlinear Kuramoto-Sivashinsky equation to develop
an accurate finite difference approximation to its dynamics. The
analysis is based upon centre manifold theory so we are assured
that the finite difference model accurately models the dynamics
and may be constructed systematically. The theory is applied after
dividing the physical domain into small elements by introducing
insulating internal boundaries which are later removed. The
Kuramoto-Sivashinsky equation is used as an example to show how
holistic finite differences may be applied to fourth order,
nonlinear, spatio-temporal dynamical systems. This novel centre
manifold approach is holistic in the sense that it treats the
dynamical equations as a whole, not just as the sum of separate
terms.