TABLE OF CONTENTS Finite-difference methods provide us with a powerful tool for generating numerical solutions to the partial differential equations of mathematical physics including the equations of fluid flow. Before, however, we can apply these methods to solve problems in fluid dynamics we require reliable and accurate strategies for computing numerical solutions to the convection diffusion equation shown in Eq. (12.1.1). The development of such methods and the investigation of their performance will be the theme of the present chapter.
The subject of finite-difference methods is broad and diverse, and we must necessarily confine our attention to discussing the fundamental principles and procedures, and presenting a selected class of methods that either illustrate the methodology or find extensive applications. Extended discussions can be found in specialized monographs and texts on numerical methods for partial differential equations including those by Richtmyer and Morton (1967), Mitchell (1969), Ames (1977), Mitchell and Griffiths (1980), Ferziger (1981), Sod (1985), Fletcher (1988, vol. I), Hirsch (1988), Hoffman (1992), and Hoffmann and Chiang (1993).
It is helpful to keep in mind throughout the present discussion that the particular way in which the convection diffusion equation enters a numerical procedure for computing the structure of a steady flow or the evolution of an unsteady incompressible flow depends upon the chosen computational strategy. In certain cases, the convection diffusion equation is integrated with reference to the equation of motion, whereas in other cases it is integrated with reference to the vorticity transport equation. Examples in each category will be discussed in Chapter 13.
