0

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.

Matthew Cassell
  • 4,248
  • 4
  • 21
  • 30
Danny
  • 877
  • 10
  • 22

1 Answers1

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.

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

Lutz Lehmann
  • 126,666
  • how it`s consistent ? please explain it to me ... – Danny Oct 11 '19 at 02:08
  • Because it has order 3. Check your definitions, consistent means that the method has at least order 1, which means $c_i=\sum_ja_{ij}$ and $\sum b_i=1$. – Lutz Lehmann Oct 11 '19 at 06:03
  • If we define the consistency by $\lim_{h\to0} \phi (x_n,y_n;h)= f(x_n,y_n)$ how can I prove it here ? can you at least give me how to write Taylor`s expansion? – Danny Oct 12 '19 at 10:30
  • For Runge-Kutta, one-step methods, this really comes back to the local truncation error order 1. For multi-step method you need additionally zero-order stability – Lutz Lehmann Oct 12 '19 at 12:15
  • I cannot use zero-stability here since I was asked to use Taylor`s theorem and not zero-stability – Danny Oct 12 '19 at 12:28
  • I know. Your are also expected to give a Runge-Kutta method, not a multi-step method. – Lutz Lehmann Oct 12 '19 at 15:44
  • any help please? – Danny Oct 12 '19 at 16:15