Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 4 / 2 / Solution

Let ${\ell }_j(\tau )={\prod }_{k\ne j}(\tau -c_k)/(c_j-c_k)$ be the Lagrange interpolation basis. Set $Y_i=P(t_n+c_{i}h)$ and $F_j=f(t_n+c_{j}h,Y_j)$. Since $P^{\prime }$ is a polynomial of degree at most $s-1$, the collocation conditions giveIntegrate from zero to $c_i$ and to one to obtainThese are the stage and update equations of a Runge-Kutta method with nodes $c_i$. Conversely, stages satisfying these equations reconstruct $P$ by integrating the displayed interpolation polynomial, so the equivalence works in both directions. Because ${\sum }_j{\ell }_j=1$, they also satisfy the collocation tableau row-sum identity ${\sum }_j{a}_{ij}=c_i$.