 Solution

The tableau is the three-stage Lobatto IIIA method. To check its order of a Runge-Kutta method directly, write $e=(1,1,1{)}^T$, $c=Ae=(0,1/2,1{)}^T$, and $C=\mathrm{diag}(c)$. The eight fourth-order conditions for a Runge-Kutta method evaluate toPowers of $c$ here mean componentwise powers. For example, $Ac=(0,1/8,1/2{)}^T$, $A{c}^2=(0,1/24,1/3{)}^T$, and $A^{2}c=(0,1/48,1/6{)}^T$, which make the last three checks immediate. These Butcher order conditions establish order at least four for general nonlinear ordinary differential equations.

It is not order five: part (b)'s stability function has expansionThe fifth coefficient already fails on the Dahlquist test equation. The order is exactly four.