A finite‐difference method of high‐order accuracy for the solution of three‐dimensional transient heat conduction problems

Abstract

A finite-difference method is presented for solving three-dimensional transient heat conduction problems. The method is a modification of the method of Douglas and Rachford which achieves the higher-order accuracy of a Crank-Nicholson formulation while preserving the advantages of the Douglas-Rachford method: unconditional stability and simplicity of solving the equations a t each time level. Although the method has not yet been applied, the analysis in this paper suggests that it will prove to be the most efficient method yet proposed for the numerical integration of three-dimensional transient heat conduction problems.Transient heat conduction problems in three dimensions represent a class of problems of great importance in many fields of engineering and science today. When the need for a numerical solution of such a problem arises, the requirements of computing time and of storage capacity often tax the capabilities of today's largest digital computers. Thus the need for more efficient finite-difference methods for solving such problems is an ever-constant one.The differential equation describing transient heat conduction in an isotropic medium of constant thermal conductivity and volumetric heat capacity is where the time scale has been normalized to include the thermal diffusivity. Three well-known finite-difference approximations to Equation (1) A.1.Ch.E. JournalConsider the following finite-difference approximation to Equation (1): A22(T*Z.,.k.m+l) + A2#(T'#,,*,fJIn Equation (6) the unknown values T*,,j,s,n+l appear only in the time difference and in the x-direction differerne. Thus only T' values along a row parallel to the x-axis are related by a system of simultaneous equations. In contrast to the solution of Equations ( 3 ) and ( 4 ) , Equation ( 6 ) requires the solution of many small, independent systems of simultaneous equations instead of one large system relating all T* values within the three-dimensional region. Furthermore each system is tridiagonal, and a very efficient method for solving the tridiagonal system without iteration is well known (1, 5, 6).If Equation (6)

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