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A bounded Euler-reaction Lie step preserves 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. Use Forward-Euler reaction followed by exact FV or periodic diffusion with a reaction-inward rectangle $R$ , an initial state in $R$ , and $0 \leq τ \leq τ_{R}$ from A bounded forward-Euler reaction step preserves an inward rectangle. Then every component of the output state lies in $R$ . The functional $M_{c, h}$ is unchanged when $c^{T} F = 0$ throughout $R$ , and is nonincreasing when $c^{T} F \leq 0$ throughout $R$ . The diffusion substep here is exact; no positivity assertion for a forward-Euler diffusion substep is made.

Facts & Assumptions

Given: One of the two finite diffusion discretizations, an inward rectangle, a reaction field satisfying the coordinate Lipschitz hypotheses of the Euler lemma, a state in $R$ , and $0 \leq τ \leq τ_{R}$ .

[F1]

Forward-Euler reaction followed by exact FV or periodic diffusion: the procedure applies the nodewise forward-Euler reaction map and then exact specieswise diffusion.

[F2]

A bounded forward-Euler reaction step preserves an inward rectangle: for $0 \leq τ \leq τ_{R}$ , the Euler reaction map sends $R$ into itself and conserves or dissipates $c^{T} z$ under the stated condition on $c^{T} F$ .

[F3]

Finite-volume and periodic diffusion preserve intervals and weighted mass: the exact diffusion map preserves each component interval and each species' weighted sum.

[F4]

Stoichiometric conservation and dissipation functionals: $M_{c, h} (U) = \sum_{i} m_{i} c^{T} U_{i}$ for the positive weights of the discretization.

Proof

1.1F1F2F4givenalgebra

The reaction substep in [F1] is the map in [F2]. Its output lies in $R$ at every cell/node. Multiplying the nodewise conservation equality or dissipation inequality by each positive weight $m_{i}$ and summing shows that this substep preserves or decreases $M_{c, h}$ as claimed.

2.1F1F3step 1.1algebra∎

Apply the exact diffusion substep in [F1]. By [F3] it keeps every species component in its interval and preserves every species' weighted sum. Hence it preserves $R$ and leaves the mass relation from step 1.1 unchanged. These are the claimed properties of the complete split update.

Depends on

Used by

A forward-Euler Fisher–KPP reaction step preserves its interval under a sharp simple bound Example

Dependency tree · two levels

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Sources

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