The previous chapter focused on the estimation of numerical errors and uncertainties due to the discretization. In addition to estimating the discretization error, we also desire methods for reducing it when either it is found to be too large or when the solutions are not yet in the asymptotic range and therefore the error estimates are not reliable. Applying systematic mesh refinement, although required for assessing the reliability of all discretization error estimation approaches, is not the most efficient method for reducing the discretization error. Since systematic refinement, by definition, refines by the same factor over the entire domain, it generally results in meshes with highly-refined cells or elements in regions where they are not needed. Recall that for 3-D scientific computing applications, each time the mesh is refined using grid halving (a refinement factor of two), the number of cells/elements increases by a factor of eight. Thus systematic refinement for reducing discretization error can be prohibitively expensive.

Targeted, local solution adaptation is a much better strategy for reducing the discretization error. After a discussion of factors affecting the discretization error, this chapter then addresses the two main aspects of solution adaptation:

1. methods for determining which regions should be adapted and

2. methods for accomplishing the adaption.

With the advent of faster computers, numerical simulation of physical phenomena is becoming more practical and more common. Computational prototyping is becoming a significant part of the design process for engineering systems. With ever-increasing computer performance the outlook is even brighter, and computer simulations are expected to replace expensive physical testing of design prototypes.

This book is an outgrowth of my lecture notes for a course in computational mathematics taught to first-year engineering graduate students at Stanford. The course is the third in a sequence of three quarter-courses in computational mathematics. The students are expected to have completed the first two courses in the sequence: numerical linear algebra and elementary partial differential equations. Although familiarity with linear algebra in some depth is essential, mastery of the analytical tools for the solution of partial differential equations (PDEs) is not; only familiarity with PDEs as governing equations for physical systems is desirable. There is a long tradition at Stanford of emphasizing that engineering students learn numerical analysis (as opposed to learning to run canned computer codes). I believe it is important for students to be educated about the fundamentals of numerical methods. My first lesson in numerics includes a warning to the students not to believe, at first glance, the numerical output spewed out from a computer.

In the next two chapters we develop a set of tools for discrete calculus. This chapter deals with the technique of finite differences for numerical differentiation of discrete data. We develop and discuss formulas for calculating the derivative of a smooth function, but only as defined on a discrete set of grid points x0, x1, …, xN. The data may already be tabulated or a table may have been generated from a complicated function or a process. We will focus on finite difference techniques for obtaining numerical values of the derivatives at the grid points. In Chapter 6 another more elaborate technique for numerical differentiation is introduced. Since we have learned from calculus how to differentiate any function, no matter how complicated, finite differences are seldom used for approximating the derivatives of explicit functions. This is in contrast to integration, where we frequently have to look up integrals in tables, and often solutions are not known. As will be seen in Chapters 4 and 5, the main application of finite differences is for obtaining numerical solution of differential equations.

Construction of Difference Formulas Using Taylor Series

Finite difference formulas can be easily derived from Taylor series expansions.