  I prove that a centre manifold approach to creating finite difference models
will consistently model linear dynamics as the grid spacing becomes small.
Using such tools of dynamical systems theory gives new assurances about the
quality of finite difference models under nonlinear and other perturbations on
grids with finite spacing. For example, the advection-diffusion equation is
found to be stably modelled for all advection speeds and all grid spacing. The
theorems establish an extremely good form for the artificial internal boundary
conditions that need to be introduced to apply centre manifold theory. When
numerically solving nonlinear partial differential equations, this approach can
be used to derive systematically finite difference models which automatically
have excellent characteristics. Their good performance for finite grid spacing
implies that fewer grid points may be used and consequently there will be less
difficulties with stiff rapidly decaying modes in continuum problems.
