Periodic rectangular reaction-diffusion grid

Fix $d\ge 1$, side lengths $L_j>0$, and integers $N_j\ge 3$. Put $h_j=L_j\mathrm{/}{N}_j$, ${\mathbb{T}}_L^d={\prod }_{j=1}^d(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$, and $G_h={\prod }_{j=1}^d(\mathbb{Z}\mathrm{/}{N}_j\mathbb{Z})$. Grid nodes are $x_i=((i_j+\frac{1}{2})h_j{)}_{j=1}^d$ modulo the side lengths, grid indices wrap modulo $N_j$, and $P_{h}u=(u(x_i){)}_{i\in G_h}$ denotes sampling. The cell volume is $m_h={\prod }_j{h}_j$. The positive periodic heat stencil is the unit-diffusivity specialization of the periodic operator in Fully discrete θ heat finite-difference scheme:

$$(A_h^{\mathrm{p}\mathrm{e}\mathrm{r}}v{)}_i=\sum _{j=1}^d\frac{2v_i-v_{i+e_j}-v_{i-e_j}}{h_j^2}\mathrm{.}$$

For $s$ species with diffusivities $d_r\ge 0$, the reaction-diffusion grid generator acts on species $r$ by $-d_r{A}_h^{\mathrm{p}\mathrm{e}\mathrm{r}}$. Each grid node has weight $m_h={\prod }_j{h}_j$; the stencil has zero row and column sums, so each species' weighted sum is conserved by diffusion. Its matrix representation satisfies $L_h=m_h{A}_h^{\mathrm{p}\mathrm{e}\mathrm{r}}$, with symmetric $L_h$, nonpositive off-diagonal entries and $L_h\mathbf{1}=0$.

The sign convention is that $A_h^{\mathrm{p}\mathrm{e}\mathrm{r}}$ is nonnegative and $-d_r{A}_h^{\mathrm{p}\mathrm{e}\mathrm{r}}$ generates diffusion. The exact eigenvalues of the positive stencil are supplied by Periodic heat-stencil eigenvalues and θ amplification factor. This grid is the only spatial discretization used in the periodic error theorem; no mesh-uniform claim for general finite-volume meshes is made there.