# Yee grid (2D FDTD)

The **Yee grid** is a staggered arrangement of electric and magnetic field samples used by [[fdtd-cuda-fdtd|FDTD]]. Instead of placing every component at the same position, it offsets them by half a cell. In FDTD CUDA's two-dimensional TMz simulation, the sample positions are:

$$
\begin{aligned}
E_z(i,j)&:\quad (i\Delta x,\;j\Delta y)\\
H_x(i,j)&:\quad (i\Delta x,\;(j+\tfrac12)\Delta y)\\
H_y(i,j)&:\quad ((i+\tfrac12)\Delta x,\;j\Delta y)
\end{aligned}
$$

This places each magnetic sample halfway between the electric samples used to update it. For example, the difference between $E_z(i+1,j)$ and $E_z(i,j)$ approximates the $x$ derivative at $H_y(i,j)$. The electric update then uses differences between the surrounding magnetic samples.

\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[x=2.8cm,y=2.8cm,>=stealth,font=\small]
  \draw[step=1,gray!35] (0,0) grid (2,2);
  % Ez samples sit at integer grid positions and point out of the plane.
  \foreach \i in {0,1,2} {
    \foreach \j in {0,1,2} {
      \draw[blue!75!black,fill=white] (\i,\j) circle (3pt);
      \fill[blue!75!black] (\i,\j) circle (1pt);
    }
  }
  % Hx is staggered along y and directed along x.
  \foreach \i in {0,1,2} {
    \foreach \j in {0,1} {
      \draw[->,red!75!black,thick]
        (\i-0.10,\j+0.5) -- (\i+0.10,\j+0.5);
    }
  }
  % Hy is staggered along x and directed along y.
  \foreach \i in {0,1} {
    \foreach \j in {0,1,2} {
      \draw[->,green!40!black,thick]
        (\i+0.5,\j-0.10) -- (\i+0.5,\j+0.10);
    }
  }
  \node[below left,blue!75!black] at (0,0) {$E_z(i,j)$};
  \node[below right,blue!75!black] at (1,0) {$E_z(i+1,j)$};
  \node[above left,blue!75!black] at (0,1) {$E_z(i,j+1)$};
  \node[above,red!75!black] at (0.12,0.5) {$H_x(i,j)$};
  \node[right,green!40!black] at (0.5,0.16) {$H_y(i,j)$};
  \draw[<->] (0,-0.42) -- node[fill=white] {$\Delta x$} (1,-0.42);
  \draw[<->] (-0.60,0) -- node[fill=white] {$\Delta y$} (-0.60,1);
  \draw[->] (2.30,0) -- (2.65,0) node[right] {$x$};
  \draw[->] (2.30,0) -- (2.30,0.35) node[above] {$y$};
  \node[anchor=west,align=left] at (0,-0.80)
    {$\odot\ E_z$: out of the plane\quad
     $\rightarrow\ H_x$\quad $\uparrow\ H_y$};
\end{tikzpicture}
\end{document}

The implementation stores each component in a separate array, using `j * pitch + i` to locate an element. The half-cell offsets are represented by the update equations; they do not require fractional array indices. Equal indices in different component arrays therefore refer to different physical positions.