Staggered compatible leapfrog

Inputs are real matrices $M_e\in {\mathbb{R}}^{n_e\times n_e}$ and $M_b\in {\mathbb{R}}^{n_b\times n_b}$ that are symmetric positive definite, $C\in {\mathbb{R}}^{n_b\times n_e}$, and compatible matrices $D,G$ satisfying $DC=CG=0$, with $S=0$ and $j\equiv 0$ in the finite compatible matrix system. All dimensions are nonnegative integers as in that system. Also input a real step size $h>0$, an integer horizon $N\ge 0$, and initial vectors $e_0\in {\mathbb{R}}^{n_e}$ and $b_{-1\mathrm{/}2}\in {\mathbb{R}}^{n_b}$. The matrix $M_e$ is positive definite, so each mass equation below has a unique exact solution, including the unique empty solve when $n_e=0$. For each integer $n=0,1,\dots ,N-1$, first set

$b_{n+1\mathrm{/}2}:=b_{n-1\mathrm{/}2}-hC{e}_n,$

then solve uniquely for $v_n\in {\mathbb{R}}^{n_e}$ in

$M_e{v}_n=C^T{M}_b{b}_{n+1\mathrm{/}2}$

and set $e_{n+1}:=e_n+h{v}_n$. Output the electric vectors $e_0,e_1,\dots ,e_N$ and magnetic vectors $b_{-1\mathrm{/}2},b_{1\mathrm{/}2},\dots ,b_{N-1\mathrm{/}2}$. When $N=0$, the loop makes no updates and the output consists only of the two initial vectors. Every mass solve is exact; no inexact-solve error model is part of this procedure.

Finite compatible matrix system

Set $t_n=nh$ and $t_{n+1\mathrm{/}2}=(n+1\mathrm{/}2)h$. The horizontal coordinate in the diagram is the normalized time $t\mathrm{/}h$. Electric vectors are stored at integer indices and magnetic vectors at half-integer indices. At each loop index, the algorithm computes $b_{n+1\mathrm{/}2}$ from $e_n$ and the previous magnetic vector, then computes $e_{n+1}$ from that new magnetic vector. The diagram records this indexing convention only; no proof uses it.