The solution of the ODE $$ y' = f(t,y)$$ is being seeked.
Let $u_{m}$ be the numerical solution of a one step method and $y(t_m)$ its true solution. The local error $e_{loc} $ is then defined as $$e_{loc}(t_m):= y(t_m) - u_m$$ Related to this this definition, is the local error per unit step, defined as $$e_{loc}(t+h)/h, \quad h \text{ being the step size.}$$ What is the deeper meaning of the defintion of the local error per unit step?
Any help would be greatly appreciated.
The local error measures how much error you make taking one step of length $h$. If you cut the step size in half you need to take twice as many steps to cover a given distance, so unless the local error is cut in half as well you are doing worse to take the shorter steps. Dividing by the step size normalizes this, so the local error per unit step is an approximation of the error you make while you advance the independent variable by $1$ regardless of the step size you take.