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I have this exercise that I can't solve. could someone explain me step by step how to solve it?

We consider an ordinary differential equation of the form

$x(a) = x_{0} \\ x′(t) = f (t, x(t)) t ∈ [a, b]$

Determine the real-valued parameters $c_{2}$ and $c_{3}$ such that the three-level Runge Kutta procedure $$\begin{array} {c|cccc} 0\\ c_{2} & c_{2} \\ c_{3}& 0& c_{3}\\ \hline & 0 &0 &1 \end{array} $$ has order 2.

MarcoDJ01
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    You need to satisfy order conditions like in https://math.stackexchange.com/questions/2531675/third-order-runge-kutta-method/4029232#4029232. Remove first the terms that contain zero as factor. – Lutz Lehmann Jul 28 '22 at 11:09
  • Sorry but I really don´t know how to do it. Can you quickly show me? – MarcoDJ01 Jul 29 '22 at 12:11
  • The first-order condition is trivially true, the second order condition reduces to $c_3=\frac12$, so you get some extended variant of the explicit midpoint method, the third order conditions can not be satisfied. There are no conditions that bind $c_2$, so you can take any value, like $1/2$, $1/3$ or $1/4$. One could try for an optimized stability region, but that is not demanded in the task. – Lutz Lehmann Jul 29 '22 at 15:45
  • However, I cannot understand what conditions they must satisfy. I read the pdf you sent, however I didn't understand much. Could you explain them to me? – MarcoDJ01 Jul 31 '22 at 08:25
  • Let's consider only autonomous ODE, this is quite general enough. The combination formula says that $\frac{y_{n+1}-y_n}{h}=f(z_3)$, $z_3=y_n+c_3hf(z_2)$. This is correct to order $O(h^2)$ if $z_3$ is close to $\frac{y_{n+1}-y_n}2$ to the same order. This means that $c_3=\frac12$, independent of what $c_2$ is. Of course, the error terms will be larger if $c_2$ is too far away from the interval $[0,1]$, but that was not the question here. – Lutz Lehmann Jul 31 '22 at 11:46
  • The order 1 and 2 conditions are simply the quadrature conditions, as the coefficients $b_k$ and $c_k$ must make up a quadrature method of at least the intended order of the RK integration method. In the order 3 conditions, the first one is again quadrature, while the second one could be vaguely characterized as extrapolation condition. – Lutz Lehmann Jul 31 '22 at 11:50

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