Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 2

Let $A=(a_{ij})$, $b=(b_i)$, $1=(1,1,1{)}^T$, and $c=A1$. From the displayed tableau,The quadrature moments satisfyDirect substitution into the remaining Butcher order conditions gives, for example,and all rooted-tree conditions of orders at most five are satisfied. The order-six moment already fails:The Runge-Kutta method therefore has

Algebraic stability requires $b_i\ge 0$ and positive semidefiniteness ofThe three weights are positive, but direct calculation givesA positive-semidefinite matrix cannot have a negative diagonal entry. Hence