Applications of Nonstandard Finite Difference Methods to Nonlinear Heat Transfer Problems

Abstract

This chapter presents some applications of nonstandard finite difference methods to general nonlinear heat transfer problems. Nonlinearity in heat transfer problems arises when i. Some properties in the problem are temperature dependent ii. Boundary conditions are described by nonlinear functions iii. The interface energy equation in phase change problems is nonlinear. Free convection and surface radiation are famous examples of nonlinear boundary conditions. The temperature distribution in multi phase regions is governed by various heat equations, for example in two phase region the temperature is governed by two different heat equation, one for solid phase, and the other one for liquid phase. Conservation of energy at a phase change interface usually yields to the nonlinear boundary condition. While particular problems presented in this research relates to nonlinear heat transfer in a thin finite rod, I fell that the methodology by which one solves these problems by nonstandard finite difference methods are quite general. I hope that these bits and pieces will be taken as both a response to a specific problem and a general method. It is common to model the nonlinear heat transfer problems by parabolic time dependent nonlinear partial differential equations. The model assumes a certain separation between time and space which is present in parabolic heat transfer problems. There are two approaches for the solution of nonlinear parabolic PDEs. One is the closed analytical form solution based on the separation of variables. Although the method of separation of variables has wide applicability, it is most often limited to linear problems. The second approach is numerical approach based on the discretization of the region. There are various approaches for performing the discretization, such as finite difference (M. Necati Ozisik et al., 1994; D. R. Croft & D. G. Lilly, 1977), finite element (R. W. Lewis, et al., 2005; J. M. Bergheau, 2010), finite volume (S. V. Patankar, 1980; H. K. Versteeg & W. Malalasekera, 1996), and spectral methods (G. Ben-Yu, 1998; O. P. Le Maitre & O. M. Knio, 2010). Here we concentrate on both standard and nonstandard finite difference methods. Standard FDs are usually used for linear part of the problem while the nonstandard FDs are used to deal with nonlinearity. Since corresponding PDEs in the heat transformation are some time dependent equations, in general one need to consider both discretizations in time and in the space. There are two typical discretization techniques in the literature. First is the semi-