Education

Abstract

Euler's method appears early and often in the undergraduate curriculum, not only as a useful method for approximating solutions to differential equations, but also as a scaffold for introducing students to the notion of consistency and accuracy. Interestingly, most students walk away with the impression that Euler, while simple and of first order, is bulletproof in the sense that as long as one is willing to wait, choosing a very small timestep will lead to a reasonable approximation of the solution. Authors Corless and Jankowski douse that notion using a leaky bucket of water in this issue of the Education section. Specifically, they use the relatively mild ODE describing the flow of water from a hole in a bucket as a model problem to explore the method of modified equations. In “Variations on a Theme of Euler,” the authors use their model problem to reintroduce readers to the residual and, from there, the concept of optimal backward error and the method of modified equations for the explicit (forward) Euler and implicit (backward) Euler methods. Using the method of modified equations, one can recast the iterative equation to show that the numerical scheme more closely approximates a modified equation that is different from the original governing equation on which the model is based. In other words, one can think of errors arising from numerical integration as errors that emerge in the modeling process. This article is a perfect module for an undergraduate modeling or numerical analysis course. Since a solid course in either area usually involves both subjects, this article is well aimed and suitable for an audience with broad interests.