Holistic projection of initial conditions onto a finite difference approximation Original Research Article

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (pdes) [B. Garcı́a-Archilla, E.S. Titi, SIAM J. Numer. Anal. 37 (2000) 470–499]. However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretization of pdes such as Burgersʹ equation [A.J. Roberts, Appl. Numer. Modelling 37 (2001) 371–396] and the Kuramoto–Sivashinsky equation [T. Mackenzie, A.J. Roberts, ANZIAM J. 42 (E) (2000) C918–C935]. In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretizations [A.J. Roberts, Comput. Phys. Comm. 126 (3) (2000) 187–206]. 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.