Variations on a Theme of Euler

This paper is an exploration of numerical methods for solving initial-value problems for ordinary differential equations. We look in detail at one “simple” problem, namely, the leaky bucket, and use it to explore the notions of explicit marching methods vs. implicit marching methods, interpolation, backward error, and stiffness. While the leaky bucket example is very well known, and these topics are studied in a great many textbooks, we will here emphasize backward error in a way that might be new and that we hope will be useful for your students. Indeed, the paper is intended to be a resource for students themselves. We will also use two techniques not normally seen in a first course, a new one, namely, “optimal backward error,” and an old one, namely, “the method of modified equations.”