[Hitchhiker 42]

Originally Posted by Richard Wallace

Hi Russ!
Should we expect, in general, that Euler integration will be well suited to approximate monotonic systems, while Midpoint integration gives better results for periodic systems? If so, why?

As I understand it, the main difference between the Euler method and the Midpoint method is that the midpoint method takes the slope by connecting a point behind and a point ahead of the given point. The Euler method just takes the slope at that point at extrapolates for that step. Please, do correct me if I'm wrong.

So, a monotonic function only increases (or decreases). It seems Euler integration would be better suited (read: more accurate) because the midpoint method depends on points behind the given point that's being calculated, which will tend to keep the slope smaller than it should be, whereas because the function doesn't tend to change direction (up or down) as much (it can only go one direction - monotonic), approximating ahead will tend to be better than approximating while taking into account behind the current point as well.