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Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me

Write the Butcher Tableau for the 1-stage $\theta$ method:

$$U^n -U^{n-1}=\tau f(\theta t_{n-1}+(1- \theta)t_n,\theta U^{n-1}+(1- \theta)U^n)$$

this is what i have attempted:

$$ U^{n+1} = U^n + \tau f(\theta t_n + (1-\theta)t_{n+1}, \theta U^n+(1- \theta)U^{n+1}) $$

substituting: $t_{n+1}=t_n+\tau$

$$ \Rightarrow U^{n+1} = U^n + \tau f(t_n + \tau(1-\theta), \theta U^n+(1- \theta)U^{n+1}) $$

from this i can get:

$$ \begin{array}{c|ccccc} 0 & 0 & 0\\ ? & ? & 0\\ --&--&--\\ & 0 & ? & \ \end{array} $$

am i along the right lines?

thanks for any help in advance.

Rodrigo de Azevedo
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user54511
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1 Answers1

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You made a good start, but I think you missed a $\theta$ in your last formula.

When writing down the Butcher tableau, remember that the number of stages in a Runge-Kutta method equals the number of times the method evaluates the function $f$. Since your method evaluates $f$ only once, it has only one stage and the Butcher tableau should have only one row above the horizontal line.

Jitse Niesen
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  • thank you very much for you help @Jitse. i think i have corrected my last equation and would $c=1-\theta$? still not quite sure how to get the other entries in tableau. thanks so much for your help again – user54511 May 01 '13 at 15:09
  • Yes, $c=1-\theta$. To get the other entries, write down the Runge-Kutta method corresponding to a Butcher tableau with one stage and $c=1-\theta$ and compare with the method you are considering. – Jitse Niesen May 01 '13 at 16:00
  • sorry- i just dont know, i feel like im going round in circles, is it ok if you tell me what the remaining elements of the table are and then im sure ill be able to work it out then. many thanks for your help it is much appreciated. – user54511 May 02 '13 at 16:32
  • does $a=1- \theta$? – user54511 May 09 '13 at 09:55
  • and does b = 1? – user54511 May 09 '13 at 10:44
  • @user54511 that's correct – Jitse Niesen May 10 '13 at 08:17