consistency of RK method

Question

I have this question which I think I don't really understand well.

Consider the IVP

\begin{align} y'(x) &= f(x,y), \quad x \in [x_0 , x_M] \\ y(x_0) &= y_0 \end{align}

Give an example of a consistent $\mathcal{O}(h^3)$ accurate three-stage RK method. Justify your answer.

I really don't understand what should I do here. Can anyone please help.

Thank you in advance.

Answer 1

Per https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/low-order-RK.pdf, the third order conditions found 1901 by W.Kutta, are

\begin{align} &&b_1+b_2+b_3&=1\\ && b_2c_2+b_3c_3&=\frac12\\ && b_2c_2^2+b_3c_3^2&=\frac13\\ \text{ and }&& b_3a_{32}c_2&=\frac16 \end{align}

Any solution gives a valid 3rd order method \begin{array}{c|ccc} 0\\ c_2&a_{21}\\ c_3&a_{31}&a_{32}\\ \hline &b_1&b_2&b_3 \end{array} explicitly named are the RK3 method and Heun's 3rd order method.

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For a demonstration of the order (and numerical convergence) of a third order method see https://math.stackexchange.com/a/3058957/115115