A characteristic‐based finite analytic method for solving the two‐dimensional steady state advection‐diffusion equation

Abstract

This paper develops an improved finite analytic (FA) solution method to the advection‐diffusion equation (ADE) for solving advection‐dominated steady state transport problems. The FA method solves the ADE analytically in localized, discrete elements, with each element linked through local boundary conditions. Previous FA methods have suffered from complex solution formulations as well as from numerical dispersion stemming from inaccuracies in the local boundary estimations. Here we use finite difference approximations of the dispersion terms to reduce solution complexity, implement an improved particle‐tracking scheme to account for velocity variations within each element, and use a Hermite interpolation scheme to estimate the local boundary conditions. The new FA method, previous FA methods, and a finite difference upwinding method are compared in both homogeneous and heterogeneous velocity fields at different Peclet numbers. Results show the new FA method exhibits little numerical dispersion without undue complexity or computational effort across all tested flow conditions.