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This problem follows from here

Consider the family of linear one-step methods defined by $$y_n = y_{n-1} + h(\theta f_n + (1 - \theta)f_{n-1})$$ where $0\leq \theta \leq 1$.

4) Consider the test problem $y' = \lambda y$, $y(0) = C$. A method is called $\delta$- damping if $$\lim_{\mathcal{R}(h\lambda)\rightarrow \infty}\frac{|y_n|}{|y_{n-1}|}$$ with $0\leq \delta < 1$. For the values of $\theta$ that yield a convergent, write conditions in terms of $\theta$ and $\delta$ that gurantee the methods is $\delta$-damping.

I am not sure how to approach this problem, any suggestions is greatly appreciated.

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Wolfy
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1 Answers1

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Just insert $$ f_k=λy_k $$ to get $$ y_n·(1-θλh)=y_{n−1}·(1+(1−θ)λh). $$ Then $$ \frac{y_n}{y_{n-1}}=\frac{1+(1−θ)λh}{1-θλh}\to -\frac{1−θ}{θ} $$ for $Re(λh)\to-\infty$.

Lutz Lehmann
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