# The limit of infinite ODE solver iteration with zero time step

Suppose I am trying to find a solution of an ordinary differential equation: \begin{aligned} y'(x) &= f(y(x))\\ y(0) &= y_0 \end{aligned} on interval $$x \in [0, x_{end}]$$ with a general linear method: \begin{aligned} Y_i &=h\sum_{j=1}^s a_{ij}f(Y_j)+\sum_{j=1}^ru_{ij}y^{[n−1]}_j,\quad i=1,2,\dots,s\\ y^{[n]}_i&=h\sum_{j=1}^sb_{ij}f(Y+j)+\sum_{j=1}^rv_{ij}y^{[n−1]}_j,\quad i=1,2,\dots,r \end{aligned} Let $$n \in \lbrace1,2,\dots,N\rbrace$$ and $$h = \frac{x_{end}}{N}$$. I am trying to find necessary and sufficient conditions under which the following limit holds: $$$$\lim_{x_{end}\rightarrow0}\lim_{N\rightarrow\infty}y_i^{[n]} = y_0$$$$ holds.

I suppose that one of consistency, zero-stability, or both (convergence) is a necessary, and/or sufficient condition for the above limit. Could you point me to which one?

p.s. A background of my questions is that I am trying to initialize a system of ODEs (multi-method integration) by fixed point iteration. I am trying to find out whether this is safe in a general case.

• I think that $$\int\frac{dy}{f(y)}=x+c$$ might be as close to a solution as you can get – clathratus Oct 28 '18 at 19:37
• This question isn't clear to me since: 1. You haven't stated how you will initialize all the required values. 2. You haven't stated how you will take the two limits (for a fixed $h$ the inner limit may not exist even if the method is convergent, so this is important). – David Ketcheson Dec 11 '18 at 2:18