 Solution

The coefficients are those of the three-stage Lobatto IIIA method. Set $e=(1,1,1{)}^T$, $c=Ae=(0,1/2,1{)}^T$ and $C=\mathrm{diag}(c)$. Direct calculation gives all eight fourth-order conditions for a Runge-Kutta method:Powers of $c$ are componentwise. These Butcher order conditions prove order at least four for a smooth general nonlinear vector field; matching a scalar test equation alone would not suffice. A necessary fifth-order condition fails:Hence the method has order exactly four, with one-step local truncation error $O(h^5)$ and, under stability and smoothness hypotheses, global error $O(h^4)$.