In order to numerically solve the ordinary differential equation $\frac{d y}{d t}=-y$ for $t>0$, with an initial condition $y(0)=1$, the following scheme is employed
\[
\frac{y_{n+1}-y_{n}}{\Delta t}=-\frac{1}{2}\left(y_{n+1}+y_{n}\right) .
\]
Here, $\Delta t$ is the time step and $y_{n}=y(n \Delta t)$ for $n=0,1,2, \ldots$ This numerical scheme will yield a solution with non-physical oscillations for $\Delta t>h$. The value of $h$ is
- $\frac{1}{2}$
- $1$
- $\frac{3}{2}$
- $2$