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Consider the initial value problem for the ODE \begin{align} \frac{dy}{dt}&=f(y), \\ y(0)&=y_0, \end{align} where $f$ is a Lipschitz continuous function on $\mathbb{R}.$ Since $f$ is globally Lipschitz, the IVP admits a unique solution globally on $\mathbb R.$

Now, consider the Euler approximation given by \begin{align} y^{n+1}=y^ n+ \Delta t f(y^n). \end{align}

For any $T>0$ and a $\Delta t > 0,$ with $N_0 \Delta t=T,$ how to prove the following estimate: \begin{align} \left|y(T)-y^{N_0}\right| \leq C \Delta t, \end{align} where $C>0$ and is independent of $\Delta t.$

P.S.: I am looking for a very general proof, i.e, for any time $T>0$ and with minimal restrictions on $f.$ I would greatly appreciate a clean proof or a precise reference that contains the proof.

Veronica
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    Possibly duplicate of https://math.stackexchange.com/questions/1921554/local-vs-global-truncation-error, https://math.stackexchange.com/questions/2365839/eulers-method-global-error-how-to – Lutz Lehmann Jul 14 '23 at 21:07
  • Well-not exactly..They assume $f''$ is Lipschitz..I am looking for the estimates when $f$ is only Lipschitz – Rosy Jul 16 '23 at 06:31
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    It's confusing. What are $y^{n+1}, y_n, y(T)$? function? value? – NN2 Jul 17 '23 at 09:09
  • I think you are actually asking to show that the global-error for the Euler-method is of order $\Delta t$. You can prove this using Taylor-expansions (assuming differentiability, not sure how many times differentiable, of $f$). – Anton Odina Jul 17 '23 at 15:19
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    @Rosy : No, only the local Lipschitz constant of $f$ in $y$-direction is used, in a region that contains the exact and numerical solution. Of the higher derivatives only their boundedness is used, that is, their continuity which implies bounds over compact sets. – Lutz Lehmann Jul 17 '23 at 15:22
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    To get global error order one, one needs that the local defect is $O(Δt^2)$, with a uniform bound. This is easiest but not most general if $y''$ is continuous, that means $f\in C^1$, at least piecewise. One could try to get along with $f$ being Lipschitz also in time direction. – Lutz Lehmann Jul 17 '23 at 18:37
  • @LutzLehmann Thanks for your comments.. But I could not follow you completely..could you please give a detailed answer – Veronica Jul 18 '23 at 16:05

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