Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 71 / 2 / b / Solution

The Runge-Kutta method has update $y_{n+1}=y_n+h{\sum }_i{b}_i{k}_i$. Expanding its squared Euclidean norm gives the exact identityThe stated per-step relation makes this difference zero. Induction from the initial vector then provesFor a positive step size the relation is also necessary for exact per-step norm conservation. This algebraic identity applies whenever the stage equations define a step; it does not silently assume solvability of arbitrary implicit stages.