I'm just curious about the following $\begin{align} H_z\big|_{i+1/2,j+1/2}^{n+1/2} = H_z\big|_{i+1/2,j+1/2}^{n-1/2} + \frac{\Delta t}{\mu_{i+1/2,j+1/2}}\Bigg[ \frac{ E_x\big|_{i+1/2,j+1}^{n} -E_x\big|_{i+1/2,j}^{n} }{\Delta y} - \frac{ E_y\big|_{i+1,j+1/2}^{n} -E_y\big|_{i,j+1/2}^{n} }{\Delta x} \Bigg] \\\\ E_x\big|_{i+1/2,j}^{n+1} = E_x\big|_{i+1/2,j}^{n} + \frac{\Delta t}{\epsilon_{i+1/2,j} \Delta y}\Bigg[ H_z\big|_{i+1/2,j+1/2}^{n+1/2} -H_z\big|_{i+1/2,j-1/2}^{n+1/2} \Bigg] \\\\ E_y\big|_{i,j+1/2}^{n+1} = E_y\big|_{i,j+1/2}^{n} - \frac{\Delta t}{\epsilon_{i,j+1/2} \Delta x}\Bigg[ H_z\big|_{i+1/2,j+1/2}^{n+1/2} -H_z\big|_{i-1/2,j+1/2}^{n+1/2} \Bigg] \end{align}$
I'm focusing on the last two finite difference updates. My question has to do for example the $E_x$ update. We observe that when $j=0$ then we need $H_z\big|_{i+1/2,-1/2}^{n+1/2}$, which is clearly outside the stencil. I'm curious how one goes about approximating it outside the domain? I have a similar issue obviously when $i=0$ with $E_y$.