Lossless or midpoint-conductive Yee leapfrog update

Choose a finite complex and boundary closure as in Oriented Maxwell cochains, masses, and the Yee grid, step size $\mathrm{\Delta }t>0$, initial $e^0\in {\mathbb{R}}^{N_e}$ and $b^{-1\mathrm{/}2}\in {\mathbb{R}}^{N_f}$, and source rows $j^{n+1\mathrm{/}2}\in {\mathbb{R}}^{N_e}$. The output is a sequence $e^n$ at integer times and $b^{n+1\mathrm{/}2}$ at half-integer times. For each $n\ge 0$, perform the ordered instructions

1. Update the face flux by $b^{n+1\mathrm{/}2}=b^{n-1\mathrm{/}2}-\mathrm{\Delta }tC{e}^n\mathrm{.}$

2. Solve the electric row $(M_e+\frac{\mathrm{\Delta }t}{2}S)e^{n+1}=(M_e-\frac{\mathrm{\Delta }t}{2}S)e^n+\mathrm{\Delta }t{C}^T{M}_b{b}^{n+1\mathrm{/}2}-\mathrm{\Delta }t{j}^{n+1\mathrm{/}2}\mathrm{.}$

For $S=0$ this is the lossless Yee leapfrog scheme. For $S⪰0$, the midpoint term is the specified trapezoidal treatment of conductivity. The row is well defined for every input because its matrix is invertible by The midpoint-conductivity Yee electric solve is well defined. This procedure makes no stability claim; the lossless stability condition is proved separately.