This book has been in preparation for over a decade. Hassan Aref and I had been making substantial additions and revisions each year, in our desire to reach the perfect book for a first course in Computational Fluid Dynamics (CFD). I sincerely wish that we had completed the book a few years ago, so that Hassan was there when the book was published. Unfortunately this was not the case. September 9th 2011 was a tragic day for fluid mechanics. We lost an intellectual leader, a fearless pioneer and for me an inspirational mentor. It is quite amazing how my academic career intersected with Hassan's over the years. He was my Masters Thesis advisor at Brown University. But when Hassan left for the University of California at San Diego, I decided to stay and finish my PhD at Brown. A few years later, when I was an assistant professor at the Theoretical and Applied Mechanics Department at the University of Illinois at Urbana– Champaign, he was appointed as the head of the department. This is when he asked me to join him in this project of writing a non-traditional introductory book on CFD. The project was delayed when Hassan moved to Virginia Tech as the Dean and I moved to the University of Florida as the Chair of the Mechanical and Aerospace Engineering Department. His passing away a few years ago made me all the more determined to finish the book as a way to honor his memory. I am very glad that the book is now finished and can be a monument to his far-sighted vision. Decades ago when CFD was in its infancy he foresaw how powerful computers and numerical methods would dominate the field of fluid mechanics.

Hassan and I shared a vision for this book. Our objective was to write something that would introduce CFD from the perspective of exploring and understanding the fascinating aspects of fluid flows. We wanted to target senior undergraduate students and beginning graduate students. We envisioned the student to have already taken a first level course in fluid mechanics and to be familiar with the mathematical foundations of differential equations.

In the previous chapter we considered initial value ODEs, where the interest was in the computation of time evolution of one or more variables given their starting value at some initial time. There is no inherent upper time limit in integrating these initial value ODEs. Therefore numerical methods for their solution must be capable of accurate and stable long time integration. By contrast, in the case of two-point boundary value and eigenvalue problems for ODEs arising in fluid mechanics, the independent space variable has two well-defined end points with boundary conditions specified at both ends. The spatial domain between the two boundary points can be infinite, as in the case of Blasius boundary layer: see (1.6.5), where the spatial domain extends from the wall (η = 0) out to infinity (η → ∞). For such a problem it is possible to treat the space variable much like time in an initial value problem, and proceed with integration from one boundary to the other and then subsequently verify the boundary conditions at the other end. We shall consider numerical methods of this sort in the next chapter.

An alternative approach is to discretize the entire domain between the two boundaries into a finite number of grid points and to approximate the dependent variables by their grid point values. This leads to a system of equations that can be solved simultaneously. Much like the time integration error considered in the previous chapter, here one encounters a discretization error. The discretization error arises from several sources: interpolation errors arise from approximating the function between grid points; differentiation errors arise in the approximation of first-, second- and higher-order derivatives; and integration errors arise from the numerical integration of a function based on its discretized values at the grid points. These errors are indeed interrelated and depend on the discretization scheme. This chapter will consider various discretization schemes. In particular, discrete approximations to the first- and second-derivative operators will be obtained. Errors arising from the different discretization schemes will be considered. The concept of discrete approximation to the first and second derivatives as matrix operators will be introduced. Finally, we will consider spatial discretization as a means to numerically integrate functions. The actual solution methodologies for two-point boundary value and eigenvalue problems for ODEs, using the tools developed in this chapter, are treated in Chapter 5.

We will begin this chapter with a discussion of two-point boundary value ODEs and then move on to two-point eigenvalue problems. The Blasius equation in Example 1.6.4 is one of the most celebrated two-point boundary value problems in fluid dynamics. Two-point boundary value problems form a challenging class of numerical computations with a wide literature. For an exposition of the theoretical ideas on two-point boundary value problems see Press et al. (1986), Chapter 16, or Keller (1968). Broadly speaking there are two classes of methods for such problems, shooting methods and relaxation methods. Shooting methods are related to integration schemes for initial value problems, while relaxation methods are based on discrete approximations to derivative operators. Thus, we are ready to tackle both these methods. In a shooting method you start from one end as in an initial value calculation using the actual boundary conditions specified at that boundary supplemented with a few other assumed (or guessed) ones which replace the actual boundary conditions of the ODE specified at the other end, and integrate (or “shoot”) towards the other boundary. In general when you eventually reach the other boundary, the boundary conditions there will not be satisfied (in other words you “miss” the target). Hence, you modify your guessed boundary conditions at the starting boundary (“aim” again) and integrate forward (“shoot” again). In this way you have in effect generated a mapping of initially guessed boundary conditions onto errors in matching the actual boundary conditions to be enforced at the other end point. Iterative procedures can now be invoked to converge to the desired value of the boundary conditions at the other end.

In a relaxation method you write the ODE(s) as a system of finite difference equations, guess a solution, and then iteratively improve it by using a root finding procedure for a system of coupled nonlinear equations. This can obviously be a rather formidable task. Often a good initial guess is needed to converge towards the correct answer. Press et al. (1986) give the whimsical recommendation for dealing with two-point boundary value problems to “shoot first and relax later.” As we shall see below, in most problems of interest in fluid dynamics the relaxation method is very powerful, exhibiting rapid convergence towards the correct answer. Furthermore, unlike the shooting method, the relaxation method is easily extendible to boundary value problems in multi-dimensions.