  Modern dynamical systems theory has previously had little to say about finite
difference and finite element approximations of partial differential equations
(Archilla, 1998). However, recently I have shown one way that centre manifold
theory may be used to create and support the spatial discretisation of \pde{}s
such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky
equation (MacKenzie, 2000). In this paper the geometric view of a centre
manifold is used to provide correct initial conditions for numerical
discretisations (Roberts, 1997). The derived projection of initial conditions
follows from the physical processes expressed in the PDEs and so is
appropriately conservative. This rational approach increases the accuracy of
forecasts made with finite difference models.
