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

Statement

Let $r > 0$ and use the scalar Fisher–KPP reaction $F (u) = r u (1 - u)$ on $R = [0, 1]$ , followed by exact no-flux TPFA diffusion with any diffusivity $d \geq 0$ . For a nonnegative step $τ$ , the forward-Euler reaction map preserves $[0, 1]$ for every starting value $u \in [0, 1]$ exactly when $0 \leq τ r \leq 1$ ; after that step, exact no-flux TPFA diffusion also preserves the interval. For $τ > 0$ , on any finite grid with positive cell weights, the reaction strictly increases the discrete mass if at least one cell value lies in $(0, 1)$ and leaves it unchanged if every cell value is a reaction equilibrium $0$ or $1$ . At $τ = 0$ the reaction map is the identity. This example makes no mass-conservation claim.

Facts & Assumptions

Given: The scalar reaction and rectangle in the Statement, the no-flux TPFA diffusion flow, and the Euler Lie update theorem.

[F1]

No-flux TPFA diffusion generator: the no-flux TPFA diffusion generator is a finite conservative diffusion operator with positive cell weights.

[F2]

Forward-Euler reaction followed by exact FV or periodic diffusion: the Euler Lie update applies $u \mapsto u + τ F (u)$ at each cell before applying exact diffusion.

[F3]

A bounded Euler-reaction Lie step preserves rectangles and mass balance: when the Euler reaction map preserves $R$ under its step restriction, exact TPFA diffusion preserves the same component interval.

[A1]

The reaction equation is the scalar Fisher–KPP field $F (u) = r u (1 - u)$ with $r > 0$ .

Proof

1.1F1A1algebra

On $[0, 1]$ , $F (0) = F (1) = 0$ , so the field points inward at both endpoints. Also $∣ F^{'} (u) ∣ = ∣ r (1 - 2 u) ∣ \leq r$ , giving the coordinate Lipschitz constant $L = r$ and Euler restriction $τ \leq 1 / r$ .

2.1A1step 1.1algebra

For $u \in [0, 1]$ , set $α = τ r \in [0, 1]$ . The Euler value is $u^{+} = u + α u (1 - u) \geq 0$ , while $1 - u^{+} = (1 - u) (1 - α u) \geq 0$ because both factors are nonnegative. Thus the reaction step preserves $[0, 1]$ .

3.1A1step 2.1choosealgebra

If $α = τ r > 1$ , choose $u$ with $1 / α < u < 1$ . Then $1 - u^{+} = (1 - u) (1 - α u) < 0$ , so the Euler map leaves $[0, 1]$ . Together with step 2.1 this proves the stated necessary-and-sufficient bound for the reaction map.

4.1F1F2F3step 2.1algebra∎

The reaction map satisfies the hypothesis of [F3] by step 2.1; applying [F2] and then [F3] proves that the full TPFA Lie update also remains in $[0, 1]$ . For a grid state $U = (u_{i})_{i}$ with positive weights $m_{i}$ , its Euler reaction mass increment is $\sum_{i} m_{i} τ r u_{i} (1 - u_{i})$ . This is strictly positive for $τ > 0$ exactly when some $u_{i} \in (0, 1)$ , and is zero when every $u_{i} \in {0, 1}$ ; for $τ = 0$ it is zero for every state. Thus the mass claim in the Statement holds, and no mass-conservation claim is made.

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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)