embedded models using simpson's rule

Question

We are looking at embedded models with Simpon's rule (order $s=4$) as primary method. We take $\hat{s}=2$. Which methods are suited as secondary models? And why? I do not understand how to choose such secondary models. And why is it not possible to find such models when we choose $\hat{s} = 3$?

Answer 1

To find suited methods, it's beneficial that the nodes used in the secondary method already be used in the primary method. This way, it won't be necessary to evaluate any new nodes which is costly for bigger problems. Given a dissection $a=x_0<x_1<\ldots<x_m=b$ of $[a,b]$ where $m\in\mathbb{N}$, what nodes does the Simpson rule use? Do you know of any other methods with order $2$ which use (part of) these nodes?