An approach to finite difference approximation is presented based on the idea of fitting the dispersion relation up to a limiting accuracy. The resulting approximations to the second derivative can be more accurate than the standard, Lagrangian finite difference approximations by an order of magnitude or more. The locality of the methods makes them well suited to parallel computation, in contrast with pseudospectral methods. The approach is illustrated with application to a simple bound state problem and to a more challenging three dimensional reactive scattering problem.