The goals of this brief chapter are to give an introduction to numerical methods other than Euler's method and to discuss error analysis in more detail. In particular, we introduce the concept of the order of a numerical method. Section 7.1 considers the errors involved in Euler's method. Sections 7.2 and 7.3 present improved Euler's method and Runge-Kutta, and Section 7.4 deals with the problem of roundoff errors arising from finite arithmetic.
Since the entire chapter deals almost exclusively with first-order equations, most of this material can be covered immediately after Section 1.4 (Euler's method). However, Sections 7.1 and 7.4 are somewhat more sophisticated than the rest of the text, so it may be appropriate to skim those sections. However, jumping to Section 7.2 without at least discussing the concept of order as it is introduced in Section 7.1 is problematic.
We expect that the students will use some type of reasonable computer technology as they solve most of the exercises in this chapter.
7.1 Numerical Error in Euler's Method
This section is a somewhat detailed discussion of the errors involved in Euler's method including an explanation of the fact that Euler's method is a first-order method. Precise error bounds are derived in Exercises 11 and 12, but the text contains a discussion of the reasons why estimates of this type are often far too conservative.
Comments on selected exercises
Exercises 1--5 study the errors involved in Euler's method as a function of step size. A fairly robust computing environment is a must for these exercises. There is an error in the back of the book in the answer to problem 1a. See errata.
Exercise 6 is a good exercise to see if the students have a practical understanding of the fact that Euler's method is a first-order method.
Exercises 7--9 use a convenient method to estimate the error in an approximation.
Exercises 10--12 are theoretical. In particular, Exercise 11 is a (long and difficult) step-by-step derivation of precise error bounds.
7.2 Improving Euler's Method
Improved Euler's method is introduced as an example of a second-order method. One goal of the section is to develop the analogy between numerical methods to approximate solutions to differential equations and numerical methods to approximate integrals. The other goal is to contrast the accuracy of a second-order method with that of a first-order method (Euler's method).
Comments on selected exercises
Exercises 1--8 of this section are the same initial-value problems as in Exercises 1--8 of Section 1.4. They compare the accuracy of improved Euler's method with that of Euler's method. The computations are possible by hand, but they would be extremely tedious.
Exercises 9--13 give students practical experience with the fact that improved Euler's method is a second-order method.
Exercises 14--17 compare the accuracy of Euler's method verses improved Euler's method.
7.3 The Runge-Kutta Method
In this section the (fourth-order) Runge-Kutta algorithm is derived. As this is a method of choice for many applications, implementations of the method are given for the TI Calculator, in Mathematica, and in the C programming language. We also use vector notation to discuss Runge-Kutta for first-order systems.
Comments on the exercises
In Exercises 1--5, we revisit initial-value problems that were studied in Sections 1.4 and 7.2. Using the results, we can compare the accuracy of Runge-Kutta with that of Euler's and improved Euler's methods.
Exercise 6 gives students practice with the fact that Runge-Kutta is a fourth-order method.
In Exercises 7 and 8, we illustrate how Runge-Kutta can be used to approximate solutions to first-order systems as well as second-order equations.
7.4 The Effects of Finite Arithmetic
The goal of this short section is to point out that, in practice, using a ridiculously small step size may not yield more accurate numerical approximations. The exercises study this phenomenon for particular examples. The answer in the back of the book to Exercise 3 is wrong; see the errata.
Comments on the Labs
Lab 7.1: Errors of Numerical Approximations
This lab relates the geometry of an equation with the accuracy of the numerical approximation. It addresses the question of why some methods are more accurate than expected on certain equations.
Lab 7.2: Lost in Space
In this lab, numerical methods are used to approximate solutions of Newton's equations. Parts of the lab could be used as detailed exercises or projects much earlier in the text.
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Version 1.1. May, 1996.