# Could anybody please clarify the relationship between numerical stability and accuracy?

I was reading a paper and came up with this statement.

Stability merely avoids uncontrolled error growth but cannot guarantee actual numerical accuracy.

From what I understood from the concept of order of accuracy, the order of accuracy is the rate of convergence of a numerical approximation of a differential equation to the exact solution.

The larger the error, the more the numerical approximation won't converge to the exact solution. Am I correct?

But why a stable method where the error is controlled cannot guarantee numerical accuracy?

This is an issue of imprecise terminology. It comes down to what you mean by "controlled". Stability* is all about the numerical solution not diverging when the exact solution isn't diverging. That means the error is controlled in the sense that it is bounded (for $h$ in some interval $(0,h_c)$ and a fixed time interval $[0,T]$, say) but not that it is going to zero as $h$ goes to zero, which is what you need for accuracy. Indeed the trivial numerical method which just doesn't do anything at all (i.e. $x_{n+1}=x_n$, regardless of what the DE says) is stable but not accurate.