Numerical calculations of pure advection based on interpolating polynomials evaluated using both the dependent variable and its derivatives are shown to be highly accurate. In one dimension, explicit methods of fourth or higher order are possible using information at only two computational points; a formal error analysis of the fourth-order method is corroborated through demonstrative calculations. Comparisons with other methods demonstrate that the two-point fourth-order method is significantly more accurate than methods of fourth or higher order which use interpolations based on four or five computational points. Extension of the principle to advection-diffusion calculations in two dimensions demonstrates excellent accuracy. The two-point higher order principle is shown to have direct applicability to the calculation of two-dimensional advection and diffusion in tidal environments at a time scale of several tidal cycles.