Two numerical integrators of the advection-diffusion equation used in air pollution simulations were tested in the framework of an evaluation methodology. The schemes were an integrator based on a polynomial approximation algorithm and a version of the pseudospectral integrator. The evaluation methodology included two test cases: the transport of a spline puff and a constant point source. Testing criteria included accuracy, conservation, and efficiency. Accuracy was assessed with the use of 'norms' which used the exact analytical solution values. The 'polynomial' integrator demonstrated superior consistency in its performance and generally better accuracy. The cost was, however, about 2.5 times that of the pseudospectral one. The pseudospectral integrator with its expanding mesh displayed deteriorating accuracy whenever the signal approached the boundaries. Simple filtering algorithms have failed to effectively control this problem.