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Sectional size discretization separates grid and cutoff errors

Statement

Let $μ_{t}$ be a continuous solution on $[0, T]$ with initial mass $M_{1}$ , and let $ν_{t}^{R}$ be the retained measure of the finite sectional system initialized by retaining the projection $P_{Δ} μ_{0}$ on pivots at most $R$ and placing the projected tail mass $\int_{(R, \infty)} x P_{Δ} μ_{0} (d x)$ in its reported reservoir. Assume $R$ is a pivot endpoint and $Δ \leq 1$ . Then $d_{B L} (ν_{t}^{R}, μ_{t}) \leq e^{C_{T} t} [C_{Q} T Δ + \frac{Δ}{2} M_{0} (μ_{t}) + \frac{M_{1}}{R} + \frac{2 K_{*} M_{1}^{2} t}{a R}], 0 \leq t \leq T,$ where $C_{Q}$ is from Fixed-pivot projection has first-order bounded-Lipschitz kernel quadrature error, $C_{T}$ is a finite bounded-number stability constant, and the terms proportional to $Δ$ , $R^{- 1}$ are respectively the grid and cutoff errors. The reservoir itself is diagnostic and does not recreate the omitted transition law.

Facts & Assumptions

Given: The continuous weak model, the finite reported-overflow scheme, and the no-cutoff pivot comparison.

[F1]

Continuous-size coagulation-fragmentation weak model: the continuous population satisfies the weak coagulation-fragmentation equation.

[F2]

Conservative fixed-pivot redistribution on a finite positive grid: in-range sizes are redistributed by positive barycentric weights and above-cutoff products enter a mass reservoir.

[F3]

Finite sectional coagulation-fragmentation balance with reported overflow: the finite algorithm applies the in-range pivot rule and omits retained particles for products above $R$ .

[F4]

Bounded-kernel measure solutions conserve mass and control number and second moments: $M_{1}$ is constant and $M_{0}$ is bounded on $[0, T]$ .

[F5]

The unbounded fixed-pivot flow has first-order grid error: the unbounded-pivot flow differs from $μ_{t}$ by at most $e^{C_{T} t} C_{Q} T Δ + (Δ / 2) M_{0} (μ_{t})$ .

[F6]

Finite cutoff differs from the unbounded pivot model by a controlled return error: replacing the unbounded-pivot flow by the finite reported-overflow flow adds at most $e^{C_{T} t} [M_{1} / R + 2 K_{*} M_{1}^{2} t / (a R)]$ .

Proof

1.1givenF1F2F3

By [F1] the target is a continuous-model solution; [F2]–[F3] specify that finite products above $R$ leave the retained measure and enter only the diagnostic reservoir. Form the unbounded-pivot solution $η_{t}$ from the initial projection $P_{Δ} μ_{0}$ . Split the difference as $d_{B L} (ν_{t}^{R}, μ_{t}) \leq d_{B L} (ν_{t}^{R}, η_{t}) + d_{B L} (η_{t}, μ_{t}) .$ This is the triangle inequality for the discrepancy in [F1].

2.1givenF4F5F6step 1.1algebra∎

Apply [F5] to the second term and [F6] to the first term. Their stated bounds use the same conserved initial mass $M_{1}$ and finite-horizon number bound, so taking a common stability constant $C_{T}$ and adding gives exactly the displayed estimate. The first two contributions vanish with grid spacing, and the last two with cutoff growth.

Depends on

Used by

Sectional forward Euler separates temporal, quadrature, and cutoff errors Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jitendra Kumar, Numerical approximations of population balance equations in particulate systems, Chapter 2, §2.2 (standard reference, not scraped)
Philippe Laurençot, Weak compactness techniques and coagulation equations, §§3.1–3.4 (standard reference, not scraped)