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.
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.