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Maxwell equations

∇⋅E=ρ,\nabla \cdot \mathbf{E} = \rho,
∇⋅B=0,\nabla \cdot \mathbf{B} = 0,
∇×E=−∂B∂t,\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},
∇×B=∂E∂t+J.\nabla \times \mathbf{B} = \frac{\partial \mathbf{E}}{\partial t} + \mathbf{J}.

Discrete version:

Eyn+1−EynΔt∣i+1/2,j,k+1/2=−Bi+1,j,k+1/2−Bi,j,k+1/2Δx∣zn+1/2+Bi+1/2,j,k+1−Bi+1/2,j,kΔz∣xn+1/2−Jy,i+1/2,j,k+1/2n+1/2,\frac{E_y^{n+1}-E_y^n}{\Delta t}\bigg|_{i+1/2, j, k+1/2} = -\frac{B_{i+1, j, k + 1/2} - B_{i, j, k + 1/2}}{\Delta x}\bigg|_z^{n+1/2} + \frac{B_{i+1/2,j, k+1} - B_{i+1/2,j, k}}{\Delta z}\bigg|_x^{n+1/2}-J_{y,i+1/2,j, k+1/2}^{n+1/2},
Bzn+1/2−Bzn−1/2Δt∣i,j,k+1/2=−Ei+1/2,j,k−Ei−1/2,j,kΔx∣yn+Ei+1/2,j+1,k−Ei+1/2,j,kΔy∣xn\frac{B_z^{n+1/2}-B_z^{n-1/2}}{\Delta t}\bigg|_{i, j, k+1/2} = -\frac{E_{i+1/2, j, k} - E_{i-1/2, j, k}}{\Delta x}\bigg|_y^{n} + \frac{E_{i+1/2,j+1, k} - E_{i+1/2,j, k}}{\Delta y}\bigg|_x^{n}