The <Butcher tableau> has one stage with $a_{11}=1/2$ and $b_1=1$, so it is the <implicit midpoint rule>
$$
y_{n+1}=y_n+h f\left(\frac{y_n+y_{n+1}}2\right).
$$
Again, interchanging the endpoints and changing $h$ to $-h$ leaves the equation invariant. Hence $\boxed{\text{method 3 is time symmetric}}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 5 / b / 3 / Solution

{c}

A Butcher tableau records a Runge-Kutta method's stage matrix $A$, nodes $c$, and output weights $b$ in the block array
$$
\begin{array}{c|c}c&A\\\hline&b^T\end{array}.
$$


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