A difference scheme for the numerical solution of an advection equation with aftereffect

We propose a family of grid methods for the numerical solution of an advection equation with a time delay in a general form. The methods are based on the idea of separating the current state and the prehistory function. We prove the convergence of the second-order method coordinatewise and do that of the first-order with respect to time. The proof is based on techniques applied for proving analogous theorems for functional differential equations and on the general theory of difference schemes. We illustrate the obtained results with a test example.