Forward-Euler reaction followed by exact FV or periodic diffusion

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 $R$; coordinate constants $L_i\ge 0$ such that $\mathrm{\mid }{F}_i(z)-F_i(w)\mathrm{\mid }\le L_i\mathrm{\mid }{z}_i-w_i\mathrm{\mid }$ whenever $z,w\in R$ differ only in coordinate $i$, as required in A bounded forward-Euler reaction step preserves an inward rectangle; ${\tau }_R={\min }_{i:L_i>0}{L}_i^{-1}$, with ${\tau }_R=+\mathrm{\infty }$ if every $L_i=0$; a state $U^n$ with every cell/node state in $R$; and $0\le \tau \le {\tau }_R$.

Procedure:

1. At every cell/node, compute the forward-Euler reaction value $V_i=U_i^n+\tau F(U_i^n)$.
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 $(V_{i,r}{)}_i$.
3. Return the resulting state $U^{n+1}$.

This is an approximate reaction substep followed by exact diffusion. The bound $\tau \le {\tau }_R$ is part of the input contract and is needed for the stated rectangle guarantee; no diffusion CFL restriction is needed because the diffusion map is exact. The procedure makes no claim about finite-precision evaluation of either step.