Holistic projection of initial conditions onto a finite difference approximation

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.Comment: 8 pages, LaTe

On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient