# Reference needed: $C^r$ convergence of Euler's method

Let $U\subset R^n$ be open, $F\colon U\to \mathbb{R}^n$ a $C^\infty$ vector field, and $x(t)$ the solution of $$x’ = F(x)$$ with initial condition $x(0) = y$, which we assume defined at least for $t\in [0,1]$. Define $x_0 = y$ and $$x^\delta_{n+1} = x^\delta_n + \delta F(x^\delta_n),$$ which corresponds to applying Euler’s method with step $\delta$. We know from Euler’s method that (if $\delta$ is small enough) $x^\delta_n - x(n\delta)$ is bounded by $C\delta$ where $C$ is a constant depending only on the bounds of the norms of $F$ and $DF$ in $U$ (the constant is something like $C_2(e^{(n+1)\delta C_1} -1)$ where $C_1, C_2$ are bounds on $F$ and $DF$ respectively). This implies that $x_n^{1/n} \to x(1)$ as $n\to \infty$.

What I can’t find in the literature is the fact that similar estimates hold for higher order derivatives with respect to the initial condition. That is, regarding $x_n^{\delta}$ and $x(t)$ as functions of the initial condition $y$, there are constants $C_k$ and $\delta_k$ depending on bounds on the norms of the derivatives of $F$ up to order $k+1$ in $U$, such that for $\delta<\delta_k$ and $n< 1/\delta$ one has $$\|D^k_y x_n^\delta - D^k_y x(n\delta)\| \leq C_k\delta.$$ The proof seems to be somewhat cumbersome. Does anyone know a reference where this is already done?

The least cumbersome way to do this is along the following lines: Take a closed neighborhood $V$ of your initial condition $y$. Then for all $z\in V$, let $x(t,z)$ be the solution of the ODE with initial condition $z$. Now view your differential equation as an abstract differential equation in the Banach space $C^r(V,R^n)$, with the initial condition being the identity function. To get what you want, you just have the extend the convergence theory for the Euler method to Banach space valued functions, which presents no major problems.

I don't know if the following references (from lecture notes on the internet) clarify enough the subject, but your question seems to be related to what is called stability of a numerical method.

http://www.math.leidenuniv.nl/~spijker/NumStab98.pdf

http://www.cems.uvm.edu/~tlakoba/math337/notes_4.pdf

Stability is related to the round-off error when calculating the numerical solution, which amounts precisely to a modification of the initial condition (and of each $x _{n}^{\delta}$) by some quantity, depending on numerical precision.

If I'm not mistaken, stability depends both on the method and the ODE to which it is applied.

In short, I don't have a precise reference, but what you are looking for seems to be the stability of Euler's method for the problem you are working on.

(posted it as an answer because it' too long for a comment)