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Characteristic-cell overlap remapping preserves mass and positivity

Statement

Assume Countable Choice and the finite measurable positive-volume partition hypotheses in Characteristic-cell overlap remapping for a conservative density. For nonnegative source averages $ρ_{j}^{n} \geq 0$ , the characteristic-cell overlap remap satisfies $ρ_{i}^{n + 1} \geq 0$ for every arrival cell and $\sum_{i} ∣ D_{i} ∣ {\overline{ρ}}_{i} = \sum_{j} ∣ C_{j} ∣ ρ_{j}^{n} .$ In particular, it satisfies $\sum_{i} ∣ A_{i} ∣ ρ_{i}^{n + 1} = \sum_{j} ∣ C_{j} ∣ ρ_{j}^{n} .$ The proof applies Overlap remapping is an L2 projection, conservative and positive to the source partition ${C_{j}}$ and departure partition ${D_{i}}$ ; its stated Countable Choice assumption is included here.

Facts & Assumptions

Given: Countable Choice, finite measurable positive-volume source and arrival partitions of the periodic box, the exact characteristic flow, and nonnegative source averages.

[F1]

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

[F2]

Overlap remapping is an L2 projection, conservative and positive: under its stated Countable Choice and finite partition hypotheses, the overlap averages are nonnegative for nonnegative source values and satisfy $\sum_{i} ∣ D_{i} ∣ {\overline{ρ}}_{i} = \sum_{j} ∣ C_{j} ∣ ρ_{j}^{n}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function; this axiom is assumed by the overlap theorem and by the local departure-volume lemma.

[F3]

A periodic C1 characteristic map is an orientation-preserving diffeomorphism: the exact flow is a diffeomorphism on the periodic lift and hence a bijection on the torus.

[F4]

Characteristic departure cells are measurable and have positive finite volume: under Countable Choice, the departure cells form a measurable partition and each has positive finite volume.

Proof

1.1givenA1F1F2F3F4algebra

By [F3] the characteristic map is bijective on the torus, and [F4] makes the departure cells a measurable positive-volume partition. Thus the source and departure partitions meet the finite-partition hypotheses of [F2]. Applying [F2] to the source values in [F1] gives ${\overline{ρ}}_{i} \geq 0$ for every $i$ and the departure-mass identity $\sum_{i} ∣ D_{i} ∣ {\overline{ρ}}_{i} = \sum_{j} ∣ C_{j} ∣ ρ_{j}^{n}$ . The Countable Choice assumption [A1] is exactly the axiom branch required by [F2] and [F4].

2.1givenF1F4step 1.1algebra∎

Since [F4] gives $∣ D_{i} ∣ > 0$ and each arrival cell has $∣ A_{i} ∣ > 0$ , [F1] scales the nonnegative ${\overline{ρ}}_{i}$ by the positive ratio $∣ D_{i} ∣ / ∣ A_{i} ∣$ , so $ρ_{i}^{n + 1} \geq 0$ . Also $∣ A_{i} ∣ ρ_{i}^{n + 1} = ∣ D_{i} ∣ {\overline{ρ}}_{i}$ term by term. Summing this equality and substituting the identity from step 1.1 proves the asserted total mass equality.

Remarks

The batch-63 definition and theorem bodies have been inspected. The theorem assumes Countable Choice and finite measurable partitions with positive cell volumes; the departure-volume lemma supplies measurability and positive volume for ${D_{i}}$ , while the periodic grid supplies those properties for ${C_{j}}$ . The cited PE17 carriers are drafts, and this Step 1 contract records mathematical and dependency readiness only; it does not claim independent approval.

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Sources

European Centre for Medium-Range Weather Forecasts, Numerical Methods, Meteorological Training Course Lecture Series, 2002 (standard reference, not scraped)
Lukas Einkemmer and Alexander Ostermann, On the error propagation of semi-Lagrange and Fourier methods for advection problems, Computers & Mathematics with Applications 69 (2015), 170–179 (standard reference, not scraped)