ON THE TIME-DISCRETIZATION METHOD FOR THE UNSTEADY HEAT CONDUCTION EQUATION WITH UNIFORM HEAT FLUX BOUNDARY CONDITION

Abstract

The time-discretization method is a powerful method to provide approximate, semi-analytic solutions of parabolic differential equations in one or more dimensions. In the case of one-dimensional parabolic partial differential equations, the time-discretization method employs the two-point backward approximation for the time-derivative, while leaving the space derivative continuous. This is a simple operation that engenders a sequence of adjoint second-order ordinary differential equations, wherein the space coordinate is the independent variable and a fixed time appears as an embedded parameter. In this work, the time-discretization method is applied to the unsteady 1-D heat equation in a large plate with constant initial temperature and uniform surface heat flux as the boundary condition. Conceptually, the associated sequence of adjoint second-order ordinary differential equations of heat conduction are of quasi-stationary nature. Using the first adjoint quasi-stationary heat equation with one time jump, it is demonstrated that an approximate, semi-analytic temperature solution of good quality is easily obtainable and is valid at all time.