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The following Runge-Kutta method is given.

$$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =\frac{5}{2} & a_{21} = \frac{5}{2} & a_{22} = 0\\ \hline & b_1 = \frac{4}{5} & b_2 = \frac{1}{5} & \ \end{array} $$

I have to determine the order of accuracy and I found that it is $2$.

Then if the method is applied at the system $\\y_1'=-80y_1+20y_2 \\ y_2'=20y_1-80y_2$ what $h$ should we take so that the calculations get done with absolute stability?

How can we find such an $ h$?

I found that the region of absolute stability is $S=\{ z \in \mathbb{C}: |\frac{z^2}{2}+z+1|<1\}$. Does this help?

evinda
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1 Answers1

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Note that the absolute stability criteria is based on applying the numerical scheme, in this case RK, to the very simple test equation $y'=\lambda y $. Assuming your derivation is correct, here is how the absolute stability region looks like in the complex plane. enter image description here. $z=h\lambda$ and you need to ensure that this is within the region plotted above. In your case, you have coupled ODEs which don't resemble the test equation, $y'=\lambda y$. A suggestion is to take $\lambda=-80$ and $\lambda=20$ respectively and check that a suitable choice of $h$ will put you in stability region.

felasfa
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