Convergence of the heat kernel approximate solutions of the transport-diffusion equation in the half-space

Abstract

A family of approximate solutions to transport-diffusion equation in the half-space R+d is constructed by using the heat kernel and the translation representing transport on each step of time discretization. The uniform convergence of these approximate solutions and that of their first derivatives with respect to the space variables as well as the point-wise convergence of their second derivatives with respect to the space variables are proved. We also show that the limit function satisfies the transport-diffusion equation in the half-space with homogenous boundary conditions. From the technical point of view it is essential to obtain for the third derivatives of approximate solutions with respect to the space variables a necessary estimate for the passage to the limit, taking into account the influence of the boundary conditions.