Section Summary

In this section we have experimented with the simple numerical approximations to initial value problems given by Euler's Method. In particular, we used this method to find approximate numerical solutions to the raindrop models of Section 5.1.

Euler's Method is not an efficient numerical scheme for solving initial value problems. We have to perform a large number of calculations to achieve accuracy. For the initial value problem

the fundamental iterative step in Euler's Method is

rise = slope \(\times\) run,

that is,

where the slope is \(f(t_{k-1},y_{k-1})\), i.e., the slope of the tangent line at the left end of the \(k\)-th interval. More sophisticated numerical methods replace this value of slope with more complicated calculations of slope.