Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 4 / 3

The collocation order theorem identifies the classical order with the order of its interpolatory quadrature. If ${\sum }_i{b}_{i}g(c_i)$ integrates every polynomial of degree at most $p-1$ exactly, but fails for some polynomial of degree $p$, then the collocation method has order $p$ for sufficiently smooth differential equations and the local implicit stage branch. With $s$ nodes, $s\le p\le 2s$.

Equivalently, put ${\pi }_s(\tau )={\prod }_i(\tau -c_i)$. If its first $k$ moments vanish,and the next does not, then $p=s+k$. Polynomial division explains the quadrature part: any polynomial through degree $s+k-1$ is a remainder of degree below $s$ plus ${\pi }_s$ times a polynomial of degree below $k$. The remainder is integrated exactly by interpolation, and the remaining integral vanishes. Conversely a nonzero next moment is an explicit failed quadrature test. The differential-equation order assertion is the collocation theorem being stated here. In particular Gauss collocation methods achieve $2s$, Radau IIA methods achieve $2s-1$, and endpoint-including Lobatto IIIA methods achieve $2s-2$ for $s\ge 2$.