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For $\theta \in [0,1]$ define the one-step theta method by $$y_{i+1} = y_i + \tau(\theta(f(t_{i+1},y_{i+1}) + (1-\theta)f(t_{i},y_{i}))$$ Find the order of consistency for the method (the answer will depend on your choice of $\theta$).

I know $\theta = 0$ and $\theta = 1$ give consistency order 1 (explicit and implicit Euler), while other choices give consistency order 2 (trapezoid method), however how can I go about proving this?

Pame
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  • This looks like a duplicate of https://math.stackexchange.com/q/1486008/115115 and https://math.stackexchange.com/q/3381289/115115. Report back if the answers there leaves questions open. – Lutz Lehmann Nov 19 '19 at 18:35
  • I saw both of those posts before asking. I don't see how they are asking the same thing. – Pame Nov 20 '19 at 08:59
  • Then please state your definition of consistency and its order. In my understanding it is done via the analysis of the local truncation error as done in those posts. You get consistency if that order is at least one. You get convergence if the method is zero-stable, which is always the case for one-step methods. – Lutz Lehmann Nov 20 '19 at 09:05

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