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A smooth deforming flow remaps two cell averages conservatively

Example

Assume Countable Choice. On the unit torus take the smooth autonomous velocity $b (x) = \sin (2 π x)$ and any fixed time $T > 0$ . Its exact flow $Φ = Φ_{T, 0}$ is an orientation-preserving diffeomorphism. Use source and arrival cells $C_{1} = A_{1} = [0, 1 / 4)$ and $C_{2} = A_{2} = [1 / 4, 1)$ in a fundamental interval, with initial cell averages $(ρ_{1}^{0}, ρ_{2}^{0}) = (1, 0)$ . The flow fixes $0$ and $1 / 2$ ; on $(0, 1 / 2)$ it moves points strictly to the right. Define $r : = Φ^{- 1} (1 / 4)$ .

Facts & Assumptions

Given: Countable Choice, the velocity, flow, two-cell partition, initial averages, and remapping formula of Characteristic-cell overlap remapping for a conservative density.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice is assumed in the Lebesgue-measurable-volume definition of the remap.

[F1]

A periodic C1 velocity has a global characteristic flow on a finite horizon: the smooth periodic velocity has a unique global flow, and constant solutions at zeros of $b$ remain fixed.

[F2]

A periodic C1 characteristic map is an orientation-preserving diffeomorphism: $Φ$ is an increasing diffeomorphism of the periodic lift and has an increasing inverse.

[F3]

Characteristic-cell overlap remapping for a conservative density: $D_{i} = Φ^{- 1} (A_{i})$ , ${\overline{ρ}}_{i} = ∣ D_{i} ∣^{- 1} \sum_{j} ∣ D_{i} \cap C_{j} ∣ ρ_{j}^{0}$ , and $ρ_{i}^{1} = (∣ D_{i} ∣ / ∣ A_{i} ∣) {\overline{ρ}}_{i}$ .

[F4]

Characteristic departure cells are measurable and have positive finite volume: under Countable Choice, the departure cells and their Lebesgue volumes are well defined and positive.

Proof

1.1givenA1F1F2F4algebra

Since $b (x) > 0$ on $(0, 1 / 2)$ , the forward trajectory from $1 / 4$ strictly increases and cannot reach the equilibrium $1 / 2$ at finite time by uniqueness in [F1]. Thus $Φ (1 / 4) > 1 / 4$ , while $Φ (0) = 0$ and $Φ (1 / 2) = 1 / 2$ . Monotonicity in [F2] gives $0 < r < 1 / 4$ . With representatives in the open fundamental interval $(0, 1)$ , $D_{1} = (0, r)$ and $D_{2} = [r, 1)$ , so $∣ D_{1} ∣ = r$ , $∣ D_{2} ∣ = 1 - r$ , and the nonzero overlaps with the occupied source cell $C_{1} \cap (0, 1)$ are $∣ D_{1} \cap C_{1} ∣ = r$ and $∣ D_{2} \cap C_{1} ∣ = 1 / 4 - r$ .

2.1givenA1F3F4step 1.1algebra∎

Formula [F3] gives ${\overline{ρ}}_{1} = 1$ and ${\overline{ρ}}_{2} = (1 / 4 - r) / (1 - r)$ . Since $∣ A_{1} ∣ = 1 / 4$ and $∣ A_{2} ∣ = 3 / 4$ , the arrival averages are $ρ_{1}^{1} = 4 r > 0$ and $ρ_{2}^{1} = \frac{4}{3} (1 / 4 - r) > 0$ . Their total mass is $\frac{1}{4} ρ_{1}^{1} + \frac{3}{4} ρ_{2}^{1} = r + (1 / 4 - r) = 1 / 4$ , equal to the initial mass $\frac{1}{4} \cdot 1 + \frac{3}{4} \cdot 0$ .

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Sources

European Centre for Medium-Range Weather Forecasts, Numerical Methods, Meteorological Training Course Lecture Series, 2002 (standard reference, not scraped)