A finite-difference scheme and a Galerkin scheme are compared with respect to a very accurate solution describing time-dependent advection and diffusion of air pollutants from a line source in an atmosphere vertically stratified and limited by an inversion layer. The accurate solution is achieved by applying the finite-difference scheme on a very refined grid with a very small time step. The grid size and time step are defined according to stability and accuracy criteria. It is demonstrated that for the problem considered the two methods are approximately equally accurate. The Galerkin method, however, gives a better approximation in the vicinity of the source. This can be assumed to be partly due to the different way the source term is taken into account in the two methods. Improvement of the accuracy of the finite difference scheme is achieved by approximating, at every step, the contribution of the source term by a Gaussian puff moving and diffusing with the velocity and diffusivity of the source location, instead of utilizing a stepwise function for the numerical approximation of \(\delta\) function representing the source term.