Localized axial Green's function method for the convection-diffusion equations in arbitrary domains

A localized axial Green's function method (LAGM) is proposed for the convection-diffusion equation. The axial Green's function method (AGM) enables us to calculate the numerical solution of a multi-dimensional problem using only one-dimensional Green's functions for the axially split differential operators. This AGM has been developed not only for the elliptic boundary value problems but also for the steady Stokes flows, however, this paper is concerned with the localization of the AGM. This localization of the method is needed for practical purpose when computing the axial Green's function, specifically for the convection-diffusion equation on a line segment that we call the local axial line. Although our focus is mainly on the convection-dominated cases in arbitrary domains, this method can solve other cases in a unified way. Numerical results show that, despite irregular types of discretization on an arbitrary domain, we can calculate the numerical solutions using the LAGM without loss of accuracy even in cases of large convection. In particular, it is also shown that randomly distributed axial lines are available in our LAGM and complicated domains are not a burden.