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Prove that the third-order Runge-Kutta method reproduces the Taylor series of the solution up to and including terms inh3for any differential equations.

$$x(t+h) =x(t) + \frac19 (2 K_1 + 3 K_2 + 4 K_3)$$

where

$$\begin{aligned} K_1 &= h f(t,x)\\ K_2 &= h f \left(t + \frac12 h, x + \frac12 K_1 \right)\\ K_3 &= h f \left(t + \frac34 h, x + \frac34 K_2 \right)\end{aligned}$$

I am really struggling with proving this. I need any help for that. Thank you.

Rodrigo de Azevedo
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Ahmed
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    So, what have you done so far? At what point are you struggling? – P. Siehr Nov 24 '17 at 13:22
  • I have found, x^(''),and x^('''), – Ahmed Nov 25 '17 at 20:26
  • I consider x^(')=f(t,x) to be any DE. Then, x^('')=(f_t+f_x .f, and x^(''')=f_tt+f_xx.f+f_x(f_t+f_x.f) where f=x^'=f(t,x). And I wanted to find K2,K3, by using a Tylor series of order 3 but I am having troble with very long term that I found? – Ahmed Nov 25 '17 at 20:31
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    Dear OP, if you are still looking for an answer, I have answered on a similar question https://math.stackexchange.com/questions/3957088/a-question-regarding-runge-kutta-method-of-order-3/3958591#3958591 – B E I R U T Feb 17 '21 at 14:40

1 Answers1

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The Butcher tableau for the given method is $$ \begin{array}{c|c}c&A\\\hline&b\end{array} \qquad=\qquad \begin{array}{c|ccc} 0&\\ \frac12&\frac12\\ \frac34&0&\frac34\\ \hline &\frac29&\frac39&\frac49 \end{array} $$ For 3-stage methods you get order conditions (see for instance Butcher 2008) \begin{align} \text{order $1$}&:& b_1+b_2+b_3&=1\\ \text{order $2$}&:& b_2c_2+b_3c_3&=\tfrac12\\ \text{order $3$}&:& b_2c_2^2+b_3c_3^2&=\tfrac13\\ & & b_3a_{32}c_2&=\tfrac16\\ \end{align}

It can be easily checked that these conditions are satisfied.

Lutz Lehmann
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