A new numerical scheme based on our propagator method is presented for solution of ADR equations. The method exploits an approach of a non-standard representation of the time derivative, by applying the derivative to the solution given as the product of two functions, where one of them is a propagator function. Propagator function is chosen in nonlocal way, and with respect to the solution, a new finite volume difference scheme is presented.
Stability of the scheme is investigated. It is shown, that stability restrictions for the propagator scheme become more weaker in comparison to traditional semi-implicit difference schemes. There are some regions of ADR coefficients, for which elaborated propagator difference scheme becomes absolutely stable. It is proven that the scheme is unconditionally monotonic. The scheme has the first order in time and the second order truncation errors in space. The scheme can be easy extended to the solution of multidimensional non-steady problems.