The Numerical Approach

Abstract

The combination of finite-difference approximations to the derivatives and the use of a high speed digital computer leads to a very powerful approach to solving the nonlinear ordinary and partial differential equations of physics. For many nonlinear systems, particularly those where the nonlinear terms are not small corrections to an otherwise linear behavior, the numerical route may be the best or only feasible way to travel. For the nonlinear ODEs encountered earlier in the text, the student has been allowed to use the Maple numerical ODE solver without any explanation provided of the principles on which it is based. In this chapter, we would like to partially fill that void by briefly describing how some of the common numerical schemes for solving nonlinear ODEs are derived. Our aim is to provide a simple conceptual framework that will make the reader more comfortable with the numerical approach while progressing through the rest of the topics that lie ahead. Since many of the underlying numerical concepts do not depend on whether the system is linear or nonlinear, occasionally a linear ODE shall be chosen for illustrative purposes because an exact analytic solution can then be generated. This will allow us to readily gauge the success, or lack thereof, of a particular numerical approach. For the same reason, the numerical schemes will be tested on nonlinear systems already solved in earlier Maple files. It should be strongly emphasized that the treatment presented in this chapter is no substitute for taking a relevant formal numerical analysis course. At best, we can only provide a very small glimpse of a vast and rapidly growing subject. It should be noted, however, that all of the concepts developed in this chapter, although expressed in terms of Maple, are relevant to programming in other computer languages.

What a chimera then is man! What a novelty! What a monster, what a chaos, what a contradiction, what a prodigy! judge of all things, feeble earthworm, depostitory of truth, a sink of uncertainty and error, the glory and the shame of the university