Exact reaction-then-diffusion Lie step on FV or periodic grids

Input: Either the no-flux TPFA discretization of No-flux TPFA diffusion generator or the periodic-grid discretization of Periodic rectangular reaction-diffusion grid; a reaction field $F$; a reaction-inward rectangle $R$ such that $F$ is $C^1$ on an open neighborhood of $R$ and $DF$ is bounded on some closed neighborhood of $R$; a state $U^n$ with each cell/node state in $R$; and a time step $\tau \ge 0$.

Procedure:

1. At every cell/node $i$, solve the exact reaction IVP $\dot{z}=F(z)$ from $z(0)=U_i^n$ to time $\tau$, and set $V_i={\mathrm{\Phi }}_F^{\tau }(U_i^n)$.
2. For each species $r$, apply the exact diffusion map $e^{-\tau d_r{M}^{-1}{L}_h}$ in the TPFA case or $e^{-\tau d_r{A}_h^{\mathrm{p}\mathrm{e}\mathrm{r}}}$ in the periodic-grid case to the vector $(V_{i,r}{)}_i$; call the output $U_{\cdot ,r}^{n+1}$.
3. Return $U^{n+1}$.

This order is the reaction-then-diffusion Lie composition. The maps in both substeps are exact flows. The algorithm makes no claim about an approximate ODE or matrix-exponential solver; any such solver error requires a separate bound. Under the hypotheses of Inward reaction fields preserve rectangles and stoichiometric mass and Finite-volume and periodic diffusion preserve intervals and weighted mass, each substep is well-defined for all finite $\tau$ on $R$.