Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 2

The Lagrange interpolation polynomial basis at $c_1,c_2$ isThe Collocation Runge-Kutta method has stages and updatewhere $a_{ij}={\int }_0^{c_i}{\ell }_j(s)ds$ and $b_j={\int }_0^1{\ell }_j(s)ds$. Explicitly,

A two-node collocation method already has order at least two. It has order at least three exactly when its node polynomial is orthogonal to constants on $[0,1]$:Equivalently,This is the additional quadrature order condition ${\sum }_i{b}_i{c}_i^2=1/3$.

Put $c_1=1/2$ and $c_2=c\ne 1/2$. The weights and stage matrix becomeFor algebraic stability of a Runge-Kutta method,soA positive-semidefinite matrix with a zero diagonal entry must have every entry in that row and column equal to zero. Here the off-diagonal entry never vanishes for finite $c\ne 1/2$. Therefore there is no admissible value of $c_2$ for which the method is algebraically stable.