A unified subroutine for the solution of 2-D and 3-D axisymmetric diffusion equation

Abstract

A unified numerical statement, which composes Galerkin finite element, subdomain finite element, finite control-volume, balanced finite difference and other numerical approximations for 2-D and 3-D axisymmetric CϖT/ϖt = Δ · (KϖT) + λT + F, was derived and formulated. The numerical statement yields Galerkin finite element, subdomain finite element, and balanced finite difference approximations by specifying a single constant, η, to 2,3 and ∞ respectively (finite control-volume approximation by lumping capacitance matrix of the subdomain finite element method). Techniques are also presented to illustrate selection of input parameters to UniSub, a Fortran implementation of the unified numerical statement, to generate correct matrix systems for the governing equation and various boundary conditions. Test problems were run to check the validity of the subroutine and to compare accuracy of various numerical schemes.

UniSub is a powerful tool to compare the accuracy of various numerical schemes since it eliminates the uncertainty of effects between codes used for comparison. UniSub can also be employed in a Fortran program to produce different numerical schemes by varying η both spatially and temporarily to achieve optimal accuracy in solving diffusion equation.