FDTD: Simulating Maxwell's Equations on a Yee Grid

Maxwell's curl equations, discretised

For a 2D transverse-magnetic wave with field components E_z, H_x and H_y, Maxwell's curl equations reduce to three coupled partial differential equations. The Finite-Difference Time-Domain method (FDTD), introduced by Kane Yee in 1966, replaces every derivative with a finite difference on a grid, and simply steps those equations forward in time — no Fourier transforms, no assumed geometry, just the local update rule applied everywhere at once:

∂H_x/∂t = -(1/μ) * ∂E_z/∂y
∂H_y/∂t =  (1/μ) * ∂E_z/∂x
∂E_z/∂t =  (1/ε) * (∂H_y/∂x - ∂H_x/∂y)

The Yee grid: why it stays accurate for free

Yee's trick is to place E and H components at offset positions: H_x and H_y sit half a cell away from E_z in space, and are also evaluated half a time step apart from it — a leapfrog in time as well as in space. Every derivative then becomes a simple, symmetric central difference, which is second-order accurate without any extra work:

H_x[i,j]^(n+1/2) = H_x[i,j]^(n-1/2) - (dt/(μ*dy)) * (E_z[i,j+1]^n - E_z[i,j]^n)
E_z[i,j]^(n+1)   = E_z[i,j]^n + (dt/ε) * ( (H_y[i,j]-H_y[i-1,j])/dx
                                                  - (H_x[i,j]-H_x[i,j-1])/dy )

The Courant stability condition

The time step cannot be chosen freely: information must not be allowed to cross more than one grid cell per step, or the scheme diverges. In two dimensions the Courant–Friedrichs–Lewy bound is:

dt ≤ dx / (c * sqrt(2))          (2D, square grid, c = speed of light in the medium)

Push the time step even slightly past this limit and the simulation does not just lose accuracy — it blows up exponentially within a handful of steps, since the numerical scheme itself becomes unstable, not merely imprecise.

Boundaries: perfect reflectors versus open space

Forcing E_z = 0 along a line of cells models a perfect electric conductor (PEC): a wave hitting it reflects completely, which is exactly how the "paint PEC reflectors" tool builds mirrors and slit walls. A simulation box that is meant to represent open space, however, cannot simply stop at its edge — a hard wall there would reflect every outgoing wave back in as spurious noise. The standard fix is a perfectly matched layer (PML), an artificial absorbing region around the edge whose impedance is engineered to match free space at every angle of incidence, so waves pass into it and are attenuated to nothing rather than bouncing back.

Diffraction and interference for free

Nothing in the update rule above knows about slits, lenses or interference — it only knows the local field values one cell away. Yet paint two narrow gaps in a PEC wall and the double-slit interference fringes appear on their own, because every point along the wavefront genuinely acts as Huygens's secondary source, exactly as the classical wave-front construction predicts. That is the real payoff of solving the field equations directly on a grid rather than hand-coding a diffraction formula: any geometry — a lens, a prism, an antenna — produces its own correct diffraction and interference pattern automatically.

Frequently asked questions

Try it live

Everything above runs in your browser — open EM Wave Simulator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.