consistency of numerical methods for ODE I was reading the Wikipedia article on numerical methods for ODEs
https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Consistency_and_order
and I saw that when it discusses "consistency and order", the consistency is defined as
$$\lim_{h\to 0} \frac{\delta^h_{n+k}}{h} = 0$$
where $\delta^h_{n+k}$ is the local truncation error with mesh size $h$.
Why is consitencty defined this way? My guess about the definition of consistency was ${\delta^h_{n+k}} \to 0$ as $h\to 0$

Suppose you have $y(0)$ and your goal is to obtain $y(t)$ where $t$ is some small positive number. If $t$ is small enough and the method is consistent, you should be able to run your numerical method with $h=t/N$ for large $N$, and then as $N \to \infty$ you should get convergence to $y(t)$. This means that you will incur the local truncation error $N=O(1/h)$ times in the course of the calculation. Thus the local truncation error had better be $o(h)$ so that the overall error is $o(1)$.
(If $t$ is no longer required to be small then you need stability as well, as you probably are aware.)
