FDM on second-order partial differential equations in 3D

We present a new technique for the approximate solution of the partial differential equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 + A 1 ∂ u ∂ x + A 2 ∂ u ∂ y + A 3 ∂ u ∂ z = 0 using finite differences. Introducing two auxiliary functions of the coordinate system and considering the form of the initial equation on lines passing through the nodal point ( x 0 , y 0 , z 0 ) and parallel to the coordinate axes, we can separate it into three parts that are finally reduced to ordinary differential equations, one for each dimension. The present technique is tested on a diffusion problem in a pipe. Our numerical results are compared to analogous results obtained with either the standard central-difference approximation or with those of the analytical solution. It is shown that the present technique is more accurate when the contribution of the first-derivative terms in the initial equation is dominant.