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 Mathematica 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. It should be emphasized that we are not attempting to explain the code which underlies Mathematica’s NDSolve command which is about 500 pages long.

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 universe.

Blaise Pascal (1623–1662)