Filter scheme methods are numerical convolution techniques for the solution of partial differential equations and are fundamentally different from finite-difference or finiteelement methods. The basic concept is that the output of any physical (linearized) system 'can be calculated from the input, if the response function is known for the system. The process becomes one of convoluting the unknown variable(s) over the space domain with a weighting function which is the Fourier transform of the response function. The method is studied for the simple one-dimensional advection problem. The raw Fourier coefficients are found analytically but require smooting by an appropriate taper function to produce a stable scheme for all wave numbers because of Gibb's effects near the Nyquist frequency. A constraining technique is also used near the very long wave scales to improve phase response of the filter. The method is shown to possess excellent numerical ability to accurately propagate very short waves associated with spiked or stepped funtions. Comparisons with various common finitedifference schemes, under conditions of equal computation effort, demonstrates the filter-scheme's superiority in this regard. The method can be extended to more general cases for which analytical solutions are not available.