On the relation between order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems

The relation between order of accuracy and convergence rate for simple linear finite difference schemes for differentiation and advection is examined theoretically and empirically. For sufficiently smooth functions, i.e. those with sufficiently steep spectral slope, the convergence rate is given by the order of accuracy. For less smooth functions, with shallower spectral slope, differentiation and advection behave slightly differently: the convergence rate of a finite difference derivative is determined entirely by the spectral slope, while the convergence rate of a finite difference advection scheme is determined by an interaction between the spectral slope and the order of accuracy.