ABSTRACT
This chapter describes the fundamentals of finite difference methods, which will be used throughout the textbook for numerical modelling of transport phenomena in materials processing. The finite difference method based on Taylor-series formulation, pioneered by Dusinberre, offers advantages in regard to numerical formulation, data preparation and computing times, but it may not be appropriate for complex geometries because of its restrictions on the element shape. The chapter discusses the finite difference methods, which are mathematically much simpler than the finite element and boundary element methods. Several procedures commonly used to develop finite difference equations include Taylor-series formulation, integral method and finite volume method. The control volume approach to derive finite difference equations has the more physical meaning compared to the use of Taylor-series expansion. Analytical methods are available with the limitation that the equations should be linear, that is, thermal and physical properties of materials involved must be regarded as independent of temperature or position.
