Approximate solutions of the advection-diffusion equation for spatially variable flows

The advection-diffusion equation (ADE) describes many important processes in hydrogeology, mechanics, geology, and biology. The equations model the transport of a passive scalar quantity in a flow. In this paper, we have developed a new approach to solve incompressible advection-diffusion equations (ADEs) with variable convective terms, which are essential to study species transport in various flow scenarios. We first reinterpret advection diffusion equations on a microscopic level and obtain stochastic differential equations governing the behavior of individual particles of the species transported by the flow. Then, simplified versions of ADEs are derived to approximate the original ADEs governing concentration evolution of species. The approximation is effectively a linearization of the spatially varying coefficient of the advective term. These simplified equations are solved analytically using the Fourier transform. We have validated this new method by comparing our results to solutions obtained from the canonical stochastic sampling method and the finite element method. This mesh-free algorithm achieves comparable accuracy to the results from discrete stochastic simulation of spatially resolved species transport in a Lagrangian frame of reference. The good consistency shows that our proposed method is efficient in simulating chemical transport in a convective flow. The proposed method is computationally efficient and quantitatively reliable, providing an alternative technique to investigate various advection-diffusion processes.