8 - Equation of Heat Transfer

Abstract

This chapter presents the equation of nonstationary heat transfer in a rod. Because of the original variable temperature distribution in the rod, heat flows from the hot regions of the rod into the cold, causing the temperature distribution to change not only along x but also with the time t. The chapter discusses finite difference approximation. Finite elements or finite differences discretize the boundary value problem by reducing the differential equation and boundary conditions to a system of linear algebraic equations. Discretization in space only of the partial differential equation reduces it to a system of ordinary differential equations in time. Finite difference discretization of the more general heat conduction problem leads to the same system of equations. The chapter presents Euler's method, which is one of the earliest and simplest of time marching schemes and is based on a backward difference formula for y. The possibilities to create higher-order explicit multistep schemes by either including more steps or higher-order derivatives of y at the nodes are enormous, but owing to stability limitations, the practical usefulness of these schemes is small. The idea behind the predictor-corrector method is to use an explicit method to predict the next y, and then to use an implicit scheme to correct it to get improved accuracy and stability.