# Truncation error and non self-starting Heun

I've seen two different truncation formulas for the midpoint rule. A common one is $$h^3 \frac{ f''}{24}$$.

Another, referred to as open Newton Cotes, is $$h^3 \frac{f''}{3}$$.

The Newton Cotes version is from a numerical method called non self-starting heun.

Why would they differ?

Vince

• Could you give a little more details? The second method could possibly compare to the first by doubling the interval length, that is, the second is over two intervals of length $h$. And what has the midpoint method to do with the trapezoidal or Heun-2 method? – Lutz Lehmann Jun 14 '19 at 21:59
• Yes, thanks, I believe doubling the length is the answer. The midpoint method is used as the predictor in the non self-starting Heun method. But instead of h+.5 as a midpoint between h and h+1, it uses h as a midpoint between h-1 and h+1. – vince Jun 15 '19 at 19:20