I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's polygonal approximations obtained by partitioning the interval $[0,T]$ in subintervals of length $\frac{T}{k}$. This procedure gives rise to a sequence of piecewise affine functions, say $\{\varphi_k\}$. Now, it is known that, under suitable hypothesis, there exists a subsequence of $\{\varphi_k\}$ that uniformly converges to a continuous function on $[0,T]$. My question is, can we prove, perhaps under further hypothesis, that the whole sequence $\{\varphi_k\}$ uniformly converges to something?
Thanks!
Yes, if the ODE satisfies the Lipschitz condition so that there exists a unique exact solution, then the Euler polygon of stepsize $h$ has an error of size $\sim \frac{M}L·(e^{Lt}-1)h$ at time $t$. $M$ is a bound on $f$ and $L$ is the Lipschitz constant of $f$.