On the approximation of the solution of transport-diffusion equation with a non-constant coefficient of diffusion

Abstract

The transport-diffusion equation with a non-constant diffusion coefficient in the whole space $\mathbb{R}^d $ is considered and a family of approximate solutions is defined by using the fundamental solution of the heat equation (heat kernel) and the translation corresponding to transport on each step of time discretization. Under appropriate conditions on the regularity of the date, the uniform convergence of approximate solutions to a function which satisfies the transport-diffusion equation is proved. To estimate and to prove the convergence of approximate solutions, we first estimate and prove the convergence of the ``positions'' with respect to which we apply the integral operator with the heat kernel. We also improve the convergence of the time derivative of approximate solutions.