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

Define the stage states $Y_i=y_n+h{\sum }_j{a}_{ij}{k}_j$, so $k_i=P{Y}_i$. Skew symmetry gives $Y_i^T{k}_i=Y_i^{T}P{Y}_i=0$, and henceInsert this into the preceding norm identity and symmetrize the double sum, using $k_i^T{k}_j=k_j^T{k}_i$. The result is the Runge-Kutta conservation of quadratic invariants identityThuswhich proves the sufficient condition for every well-defined step and then for all $n$ by induction. For example the implicit midpoint rule, with $a_{11}=1/2$ and $b_1=1$, has $M_{11}=0$ and preserves this norm exactly. The condition is sufficient, and need not be necessary for a particular matrix or a particular solution.