Upstream interpolation with a cubic space is investigated as a way of integrating the advective equation. In advection tests with a cone this is found to give much better results than realized with second-order conservative centered differencing on a double resolution mesh, and used one-third the computation time and one eighth of the memory space. The phase errors are less than those of the fourth-order Arakawa scheme at double the resolution. The integration scheme used is second-order accurate in time, and can easily he combined with can “leapfrog” approximations as a practical way of exploiting the advantages of both types of approximation for general problems. The spline interpolation representation of advection should he of use where boundary conditions are not periodic and where the exact advection of a conservation law is not as important as good phase and amplitude fidelity.