Convergence of solution of transport-diffusion system to that of transport system

In this paper we prove the convergence of the solution of transport-diffusion equation system to the solution of transport equation system in the whole d-dimentional Euclidean space when the diffusion coefficient tends to 0. In particular, it is proved that the difference between each term of transport-diffusion equation and the corresponding term of transport equation tends to 0 proportionally to the coefficient of diffusion. The proof is based on the comparison of approximate solutions for transport-diffusion equation with those for transport equation. These approximate solutions for transport-diffusion equation are constructed by the fundamental solution of diffusion equation and by translation on each step of time discretization, while the approximate solutions for transport equation are constructed by translation on the same time discretization.