# Maxwell on a Yee grid

FDTD replaces spatial curls with finite differences and advances electric and magnetic fields in a leapfrog sequence. For linear media, the continuous equations are

$$
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad
\nabla\times\mathbf H=\frac{\partial\mathbf D}{\partial t}+\mathbf J,
$$

with constitutive relations $\mathbf D=\boldsymbol\varepsilon\mathbf E$ and $\mathbf B=\boldsymbol\mu\mathbf H$. FDTDX stores the three vector components on offset locations in each cell so a discrete curl naturally connects neighboring samples.

```{figure} ../_static/generated/yee_cell.svg
:alt: Electric and magnetic field components staggered on a Yee cell

Electric components live on edges; magnetic components occupy complementary staggered locations. Material values are mapped onto the corresponding component locations.
```

## Leapfrog update

Ignoring conductivity and PML terms for the moment, one time step is conceptually

$$
\mathbf H^{n+1/2}=\mathbf H^{n-1/2}-\Delta t\,\boldsymbol\mu^{-1}\nabla_h\times\mathbf E^n,
$$

$$
\mathbf E^{n+1}=\mathbf E^n+\Delta t\,\boldsymbol\varepsilon^{-1}\left(\nabla_h\times\mathbf H^{n+1/2}-\mathbf J^{n+1/2}\right).
$$

FDTDX represents inverse material tensors because those are applied directly in the update. Conductivity, dispersive-pole state, and PML auxiliary fields add local terms but preserve the same rhythm.

## Stability is necessary, not sufficient

For a uniform 3D grid, a familiar Courant bound is

$$
\Delta t \leq \frac{S}{c\sqrt{\Delta x^{-2}+\Delta y^{-2}+\Delta z^{-2}}}, \qquad S\leq1.
$$

Staying below it prevents the ordinary Yee scheme from diverging. It does not guarantee accuracy: numerical phase velocity depends on direction and cells per wavelength. Interfaces introduce staircasing error, and resonances demand enough runtime to resolve narrow spectral structure.

## Why FDTD suits JAX

Each time step is a regular stencil over dense arrays. JAX compiles that array program for GPU, CPU, or TPU; `lax.scan`/loop constructs avoid Python overhead; and the same program participates in automatic differentiation. Static shapes are important: changing a grid dimension causes recompilation.

## Precision and scaling

Most benchmarks use `float32`, which is efficient on consumer GPUs and accurate enough when discretization error dominates. Use nondimensional diagnostics—relative energy decay, normalized mode power, $R+T$, convergence with resolution—rather than judging raw array magnitudes alone.