Asymptotic analysis of finite difference methods

If we consider a finite difference method simply as a set of equations containing a small parameter (the grid spacing), it is evident that the tools of asymptotic analysis can give us useful information about the method. The applicability of this approach for studying consistency, long time behavior and stability is demonstrated. As example, we use a simple lattice Boltzmann scheme for the 1D advection equation with constant advection velocity. Applications of the method to lattice Boltzmann schemes for the Navier-Stokes equation can be found in [1, 2, 3, 5, 6, 7]. It should be stressed that the results are not restricted to lattice Boltzmann methods but can readily be applied to any other finite difference scheme (see [4] for various examples and a short review of asymptotic methods in numerical analysis).