Convection of physical quantities of random density

We study the random flow, through a thin cylindrical tube, of a physical quantity of random density, in the presence of random sinks and sources. We model convection in terms of the expectations of the flux and density and solve the initial value problem for the resulting convection equation. We propose a difference scheme for the convection equation, that is both stable and satisfies the Courant–Friedrichs–Lewy test, and estimate the difference between the exact and approximate solutions.