The finite-difference time-domain (FDTD) method for Maxwell’s equations allows calculating the electromagnetic fields $E$ and $H$ as functions of space and time. Suppose that we have a two-dimensional space grid with points specified by $x=i\Delta x$, $y=j\Delta y$. At the point $i=10$, $j=10$ the electromagnetic field is constant: $E=E_0$, $H=H_0$ for all $t=n\Delta t$. Everywhere else $E=H=0$. From physical considerations it’s clear that this field configuration is constant and cannot propagate in space. But according to FDTD the field in the points around $i=10$, $j=10$ is defined by expressions like $(E(i=10)-E(i=9))/\Delta x$. This means we will get a propagating electromagnetic wave. How can this contradiction be explained?
If I understand correctly, $(E(i=10)-E(i=9))/\Delta x$ would be non-zero at $t=0$. But then, qualitatively, and that's enough since you said you oversimplified, that means that the rotational and divergence of $E$, and similarly of $H$, may have a non-zero value at the grid point $(9,10)$. Now look at Maxwell equations: it means the derivatives of $E$ and $H$ may have non-zero values at that grid point. Thus at the next tick $t=\Delta t$, the fields would then have a non-zero value at grid point $(9,10)$. Then rinse and repeat and the grid points will keep populate with non-zero field values.