Finite compatible leapfrog evolution

Use the finite cochain data fixed by the preceding definition item, namely finite real coordinate spaces $C^0,C^1,C^2,C^3$, maps $D_0,D_1,D_2$ with $D_1{D}_0=D_2{D}_1=0$, and symmetric positive-definite mass maps $M_1,M_2$. Inputs are rational matrices $D_0,D_1,D_2,M_1,M_2$, rational initial coordinates $b_0\in C^2$ and $d_{1\mathrm{/}2}\in (C^1{)}^{\ast }$, a positive rational step size $\tau$, and an integer $N\ge 0$. The matrices and states use the fixed ordered bases and dual bases from that definition. All arithmetic is exact over $\mathbb{Q}$; floating-point arithmetic is not part of this algorithm.

First compute $R=M_1^{-1}$ once by fixed-order, no-row-swap Gauss–Jordan elimination of the augmented matrix $[M_1\mid I_{n_1}]$. For each pivot column $j=1,\dots ,n_1$, let $p$ be the current entry in position $(j,j)$. If $p=0$, return an inconsistency error. Otherwise divide row $j$ by $p$ and, for every $k\mathrm{\ne }j$, replace row $k$ by row $k$ minus its current column-$j$ entry times row $j$. After all columns are processed, let $R$ be the right block. For $n_1=0$, take the unique empty matrix for $R$ and perform no elimination.

Set $(b,d):=(b_0,d_{1\mathrm{/}2})$ and record this initial state. For each $n=0,\dots ,N-1$, in the stated order compute

$b_{\mathrm{n}\mathrm{e}\mathrm{w}}:=b-\tau D_{1}Rd,d_{\mathrm{n}\mathrm{e}\mathrm{w}}:=d+\tau D_1^{\mathsf{T}}{M}_2{b}_{\mathrm{n}\mathrm{e}\mathrm{w}}\mathrm{.}$

Append $(b_{\mathrm{n}\mathrm{e}\mathrm{w}},d_{\mathrm{n}\mathrm{e}\mathrm{w}})$ to the output list and then set $(b,d):=(b_{\mathrm{n}\mathrm{e}\mathrm{w}},d_{\mathrm{n}\mathrm{e}\mathrm{w}})$. After the loop, return $R$ together with the list of the $N+1$ states $(b_n,d_{n+1\mathrm{/}2})$ for $0\le n\le N$. The error branch is unreachable on admissible inputs; the separate finite-step termination lemma proves this and the exact inverse output.

Finite cochain Maxwell phase space

Finite arithmetic termination and cost ↗

The update uses the state order $b_n,d_{n+1\mathrm{/}2}$ and this ordering fixes the sign of the cross term in its invariant. Positive definiteness makes every pivot in the fixed no-row-swap schedule nonzero; the separate lemma proves this by Schur-complement induction. The source’s Whitney finite-element leapfrog has its own spaces, mass matrices, boundary conditions and CFL hypotheses. This algorithm makes no claim of being its spatial discretization or of giving a continuum stability or convergence result.