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Exact reaction-diffusion Lie steps preserve rectangles and mass balance

Statement

Choose either the no-flux TPFA system of No-flux TPFA diffusion generator or the periodic grid system of Periodic rectangular reaction-diffusion grid, and use the exact reaction-then-diffusion step of Exact reaction-then-diffusion Lie step on FV or periodic grids. Let $R = \prod_{r = 1}^{m} [a_{r}, b_{r}]$ be reaction-inward, and suppose $F$ is $C^{1}$ on a neighborhood of $R$ with derivative bounded on a closed neighborhood. If every cell/node state of $U^{n}$ lies in $R$ , then every cell/node state of $U^{n + 1}$ lies in $R$ for every finite $τ \geq 0$ . If $c^{T} F = 0$ throughout $R$ , then $M_{c, h} (U^{n + 1}) = M_{c, h} (U^{n})$ ; if $c^{T} F \leq 0$ throughout $R$ , then $M_{c, h} (U^{n + 1}) \leq M_{c, h} (U^{n})$ . Here $M_{c, h} (U) = \sum_{i} m_{i} c^{T} U_{i}$ uses the cell/node weights of the chosen discretization.

Facts & Assumptions

Given: One of the two stated finite diffusion discretizations, a reaction-inward rectangle $R$ , a $C^{1}$ field with bounded derivative near $R$ , a conservation or dissipation vector $c$ , and an initial grid state in $R$ .

[F1]

Exact reaction-then-diffusion Lie step on FV or periodic grids: the step applies the exact reaction flow at each cell/node and then the exact specieswise diffusion flow.

[F2]

Inward reaction fields preserve rectangles and stoichiometric mass: the reaction flow exists for every finite nonnegative time, maps $R$ into itself, and preserves or dissipates $c^{T} z$ under the corresponding condition on $c^{T} F$ .

[F3]

Finite-volume and periodic diffusion preserve intervals and weighted mass: each specieswise diffusion flow maps its component interval into itself and preserves that species' weighted sum.

[F4]

Stoichiometric conservation and dissipation functionals: the weighted discrete functional is $M_{c, h} (U) = \sum_{i} m_{i} c^{T} U_{i}$ with positive cell/node weights $m_{i}$ .

Proof

1.1F1F2F4givenalgebra

After the reaction substep of [F1], [F2] gives $V_{i} \in R$ at every cell/node. It also gives $c^{T} V_{i} = c^{T} U_{i}^{n}$ in the conservation case or $c^{T} V_{i} \leq c^{T} U_{i}^{n}$ in the dissipation case. Multiplying by $m_{i} > 0$ and summing over the finite index set proves the corresponding mass relation for $V$ .

2.1F1F3step 1.1algebra∎

Apply the diffusion substep of [F1] to $V$ . By [F3], each species remains in its interval, so every resulting cell/node state remains in $R$ . Also [F3] preserves each species' weighted sum; taking the finite linear combination with coefficients $c_{r}$ preserves $M_{c, h}$ . Thus the mass relation from step 1.1 is unchanged by diffusion, proving both conclusions for $U^{n + 1}$ .

Depends on

Used by

Mesh-uniform first-order Lie error for a smooth periodic reaction-diffusion solution Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Willem Hundsdorfer, Numerical Solution of Advection-Diffusion-Reaction Equations, lecture notes for Ph.D. course, 1996 (standard reference, not scraped)