The one-dimensional formulation assumes that electromagnetic fields vary only along the $x$-axis. FDTD OpenMP keeps the electric component $E_z$ and magnetic component $H_y$, with all other components zero. Both retained components are transverse to the direction of propagation.
Including electric and magnetic loss terms, the equations are:
$$ \begin{aligned} \frac{\partial H_y}{\partial t} &=\frac{1}{\mu}\frac{\partial E_z}{\partial x}-\frac{\sigma_m}{\mu}H_y \\ \frac{\partial E_z}{\partial t} &=\frac{1}{\varepsilon}\frac{\partial H_y}{\partial x}-\frac{\sigma_e}{\varepsilon}E_z \end{aligned} $$
The positive spatial derivatives follow from the chosen field orientation. Electric conductivity describes electric losses; fictitious magnetic conductivity provides matched damping in the PML. Both vanish in the default vacuum interior.
This model represents plane waves passing through layers perpendicular to $x$. Material changes can alter wave speed and produce reflection or transmission. It cannot represent diffraction around an obstacle or a transverse beam profile.
The Yee grid turns these coupled equations into neighboring array accesses. Update coefficients incorporate local material properties and loss terms before stepping begins.