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Fixed-pivot weights are nonnegative and reproduce number and mass

Statement

For every $z \in [x_{i}, x_{i + 1}]$ , the two weights in Conservative fixed-pivot redistribution on a finite positive grid are nonnegative, sum to $1$ , and have weighted pivot mass $z$ . Their expected absolute displacement is at most $Δ / 2$ , and their weighted second moment satisfies

$\int u^{2} P_{Δ} (z) (d u) - z^{2} = (z - x_{i}) (x_{i + 1} - z) \leq \frac{Δ^{2}}{4} .$

At a grid endpoint the same conclusions hold for the single-pivot endpoint rule, with zero second-moment excess.

Facts & Assumptions

Given: A strictly increasing positive grid and $z \in [x_{i}, x_{i + 1}]$ .

[F1]

Conservative fixed-pivot redistribution on a finite positive grid: the weights are $(x_{i + 1} - z) / (x_{i + 1} - x_{i})$ and $(z - x_{i}) / (x_{i + 1} - x_{i})$ , with single-node assignment at grid endpoints.

Proof

1.1givenF1algebra

Since $x_{i} \leq z \leq x_{i + 1}$ and $x_{i + 1} - x_{i} > 0$ , both fractions in [F1] are nonnegative. Their sum is $((x_{i + 1} - z) + (z - x_{i})) / (x_{i + 1} - x_{i}) = 1$ .

2.1step 1.1F1algebra

Their weighted mass is $\frac{(x_{i + 1} - z) x_{i} + (z - x_{i}) x_{i + 1}}{x_{i + 1} - x_{i}} = z$ after expanding and cancelling terms.

2.2givenF1step 1.1algebra

Writing $h_{i} = x_{i + 1} - x_{i}$ , the weighted second moment is $\frac{x_{i + 1} - z}{h_{i}} x_{i}^{2} + \frac{z - x_{i}}{h_{i}} x_{i + 1}^{2}$ . Subtracting $z^{2}$ and factoring gives $(z - x_{i}) (x_{i + 1} - z)$ , which is nonnegative and at most $h_{i}^{2} / 4 \leq Δ^{2} / 4$ because $(z - x_{i}) + (x_{i + 1} - z) = h_{i}$ .

3.1givenF1step 2.1step 2.2

At a grid endpoint the single pivot has mass $z$ and second moment $z^{2}$ , so the excess is zero; this proves the endpoint clauses too.

4.1givenF1step 2.2algebra∎

In an interior cell, the expected absolute displacement is $\frac{x_{i + 1} - z}{h_{i}} (z - x_{i}) + \frac{z - x_{i}}{h_{i}} (x_{i + 1} - z) = \frac{2 (z - x_{i}) (x_{i + 1} - z)}{h_{i}} \leq \frac{h_{i}}{2} \leq \frac{Δ}{2},$ by step 2.2. At a node the displacement is zero by [F1].

Depends on

Conservative fixed-pivot redistribution on a finite positive grid

Used by

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Sources

Jitendra Kumar, Numerical approximations of population balance equations in particulate systems, Chapter 2, §2.2 (standard reference, not scraped)