It is well known that Euler approximation of an ODE $$dx(t) = f(t,x(t))dt$$ can be defined as follows, for a uniform partition of coarseness $1/N$ over $[0,1]$, $$ x_{t_{n+1}} = x_{t_n} + \frac{1}{N} f(t_n, x_{t_n}).$$
If we were to replace $1/N$ with $1/N^{1/2}$ does the Euler approximation scheme $$x_{t_{n+1}} = x_{t_n} + \frac{1}{N^{1/2}} f(t_n, x_{t_n})$$ converges in $L^2$ to the SDE $$dx(t) = f(t,x(t))dB_t$$ with $B_t$ being a brownian motion? Are there any reference to a similar result under reasonable assumptions?