PDE boundary conditions from minimum reduction of the PDE

Abstract

Convection-diffusion partial differential equations (PDEs) are second order in the direction of flow, and therefore require two boundary conditions with respect to the spatial independent variable in that direction. The entering or “inflow” boundary condition is usually obvious. The exiting or “outflow” boundary condition is not so obvious. We propose reducing the PDE by one order in the direction of flow to produce the outflow boundary condition. This procedure is illustrated with a dynamic version of the classical Graetz problem in heat transfer. The proposed procedure has the advantages of physically meaningful boundary conditions which produce numerical solutions of good accuracy, and ease of implementation in a method of lines code.