Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 72 / 4 / a / Solution

For an $s$-stage Runge-Kutta method with coefficient matrix $A=(a_{ij})$ and weights $b=(b_i)$, put $B=\mathrm{diag}(b_i)$. It is algebraically stable whenEquivalently, $m_{ij}=b_i{a}_{ij}+b_j{a}_{ji}-b_i{b}_j$ defines a positive semidefinite symmetric matrix. The condition concerns the full coefficient tableau and nonlinear contractivity, not merely the scalar stability function. It is normally used with a consistent Runge-Kutta method and stage equations satisfying the necessary stage solvability of an implicit Runge-Kutta method.