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Two-species continuous and discrete dispersion criteria

Statement

Let $F \in C^{1}$ and let $y_{*} \in R^{2}$ be a homogeneous equilibrium, $F (y_{*}) = 0$ . Set $J = D F (y_{*}) = (J_{i j})$ , assume $tr J < 0$ and $\det J > 0$ , and let $D = diag (d_{1}, d_{2})$ with $d_{1}, d_{2} > 0$ . For any spatial eigenfunction of $- Δ$ with eigenvalue $λ \geq 0$ , the linearized reaction-diffusion mode has matrix $J - λ D$ . For the continuous rectangular torus, $λ = λ_{k} = \sum_{j} (2 π k_{j} / L_{j})^{2}$ with $k \in Z^{d}$ . For the periodic grid of Periodic rectangular reaction-diffusion grid, the mode indexed by $ℓ$ has $λ = λ_{h, ℓ} = 4 \sum_{j} h_{j}^{- 2} \sin^{2} (π ℓ_{j} / N_{j})$ , with $0 \leq ℓ_{j} < N_{j}$ .

Define

β_{T} = d_{2} J_{11} + d_{1} J_{22}, q (λ) = \det (J - λ D) = d_{1} d_{2} λ^{2} - β_{T} λ + \det J .

For either domain or grid and each actual mode, $J - λ D$ has a positive-real-part eigenvalue exactly when $q (λ) < 0$ ; it is Hurwitz exactly when $q (λ) > 0$ , and has a zero eigenvalue when $q (λ) = 0$ . Thus the continuous or discrete linear instability criterion tests the actual corresponding spectrum. If all nonnegative $λ$ are regarded as continuously available, then a diffusion-driven unstable band exists exactly when $β_{T} > 0$ and $β_{T}^{2} > 4 d_{1} d_{2} \det J$ ; its endpoints are

λ_{\pm} = \frac{β_{T} \pm \sqrt{β_{T}^{2} - 4 d_{1} d_{2} \det J}}{2 d_{1} d_{2}} .

On a bounded torus or a finite grid, the band criterion alone is not enough: at least one actual Laplacian eigenvalue must lie strictly between $λ_{-}$ and $λ_{+}$ .

Facts & Assumptions

Given: The smooth two-component reaction field, homogeneous equilibrium, kinetic Jacobian $J$ , positive diagonal diffusivities, and either a continuous periodic rectangular domain or the finite periodic grid.

[F1]

Semilinear vector reaction-diffusion systems: the diffusion matrix is diagonal and the reaction term is local, so linearization at a homogeneous equilibrium gives $w_{t} = D Δ w + J w$ .

[F2]

Periodic rectangular reaction-diffusion grid: the positive centered heat stencil is $A_{h}^{p e r}$ and the diffusion generator is its negative times $D$ .

[F3]

Periodic heat-stencil eigenvalues and θ amplification factor: on a periodic rectangular grid the Fourier modes form a basis and the positive heat stencil has eigenvalues $4 \sum_{j} h_{j}^{- 2} \sin^{2} (π ℓ_{j} / N_{j})$ when the directional diffusivities are one.

Proof

1.1F1givenconstructalgebra

Write $u = y_{*} + ε w$ . Differentiability of $F$ gives $F (y_{*} + ε w) = F (y_{*}) + ε J w + o (ε) = ε J w + o (ε)$ , so the linearized equation is $w_{t} = D Δ w + J w$ . For a continuous torus mode $w (t, x) = a (t) e^{2 π i \sum_{j} k_{j} x_{j} / L_{j}}$ , direct differentiation gives $Δ w = - λ_{k} w$ and hence $a^{'} = (J - λ_{k} D) a$ .

1.2givenalgebra

For every $λ \geq 0$ , $tr (J - λ D) = tr J - λ (d_{1} + d_{2}) < 0$ . Expanding the $2 \times 2$ determinant gives $\det (J - λ D) = (J_{11} - λ d_{1}) (J_{22} - λ d_{2}) - J_{12} J_{21} = d_{1} d_{2} λ^{2} - β_{T} λ + \det J = q (λ)$ .

2.1F2F3step 1.1algebra

For the grid, insert a Fourier vector from [F3] into the linearized finite system. By [F2] its diffusion part multiplies that mode by $- D λ_{h, ℓ}$ ; the reaction Jacobian acts on the species coordinate by $J$ . Therefore the mode coefficient satisfies $a^{'} = (J - λ_{h, ℓ} D) a$ . The basis assertion in [F3] ensures these are the actual grid mode matrices.

2.2step 1.2algebra

The characteristic polynomial of a real $2 \times 2$ matrix $A$ is $μ^{2} - tr (A) μ + \det (A)$ . If $tr (A) < 0$ and $\det (A) < 0$ , its discriminant $tr (A)^{2} - 4 \det (A)$ is positive and the real roots have negative product, so one is positive. If $tr (A) < 0$ and $\det (A) > 0$ , real roots have negative sum and positive product, so both are negative; nonreal roots are conjugate with real part $tr (A) / 2 < 0$ . If the determinant is zero, one eigenvalue is zero and the other is the trace, which is negative. Applying these alternatives to $A = J - λ D$ proves the mode tests in the Statement.

3.1step 1.2step 2.2algebra∎

Since $d_{1} d_{2} > 0$ and $\det J > 0$ , the upward-opening quadratic $q$ is negative at some $λ \geq 0$ exactly when its minimum occurs at a positive point and its discriminant is strictly positive. Its minimizer is $β_{T} / (2 d_{1} d_{2})$ , so the first condition is $β_{T} > 0$ ; its discriminant is $β_{T}^{2} - 4 d_{1} d_{2} \det J$ , so the second condition is the strict inequality in the Statement. Under these conditions both roots are positive because their sum is $β_{T} / (d_{1} d_{2}) > 0$ and product is $\det J / (d_{1} d_{2}) > 0$ , and factorization gives $q < 0$ precisely between them. The mode tests from step 2.2 then prove the continuous-spectrum and finite-grid criteria without replacing either actual spectrum by all real wavenumbers.

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Sources

Luca Doria, Numerical Algorithms, lecture notes, Johannes Gutenberg-Universität Mainz, 2022 (standard reference, not scraped)