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The sectional forward-Euler step is positive and conserves augmented mass

Statement

For a nonnegative finite sectional state, if $h Λ_{i} (n) \leq 1$ for every bin, the successful step of Forward Euler for a finite sectional balance with a positivity CFL has $n_{i}^{+} \geq 0$ , $L^{+} \geq 0$ , and

$\sum_{i} x_{i} n_{i}^{+} + L^{+} = \sum_{i} x_{i} n_{i} + L .$

Facts & Assumptions

Given: A valid grid state $n_{i}, L \geq 0$ , a step $h > 0$ , and the CFL inequalities in the Statement.

[F1]

Forward Euler for a finite sectional balance with a positivity CFL: the update is $(n^{+}, L^{+}) = (n, L) + h F (n, L)$ and $Λ_{i} (n) = \sum_{j} K_{i j} n_{j} + β_{i}$ .

[F2]

Finite sectional event dynamics preserve positivity and retained-plus-reservoir mass: the augmented mass is conserved along the event ODE.

[F3]

Finite sectional coagulation-fragmentation balance with reported overflow: each ordered pair has rate $K_{i j} n_{i} n_{j} / 2$ and removes both parents; fragmentation removes its parent at rate $β_{i} n_{i}$ ; retained gains are nonnegative, and overflow adds its consumed mass to $L$ .

Proof

1.1givenF1F3algebra

For each bin, write $F_{i} (n) = G_{i} (n) - n_{i} Λ_{i} (n)$ , where $G_{i} (n) \geq 0$ is the total birth rate. This is the event decomposition of [F3]: the total aggregation loss of parent $i$ is $n_{i} \sum_{j} K_{i j} n_{j}$ under the ordered-pair convention, and its fragmentation loss is $β_{i} n_{i}$ . Thus $n_{i}^{+} = n_{i} (1 - h Λ_{i} (n)) + h G_{i} (n) \geq 0$ by the CFL condition.

2.1givenF1F2F3algebra∎

By [F3], the reservoir update adds $h$ times a nonnegative overflow-mass rate, so $L^{+} \geq 0$ . By [F2], the mass derivative at every nonnegative initial state is zero, so the vector-field increment has zero augmented mass; multiplying by $h$ and adding the unchanged initial mass proves the displayed identity.

Depends on

Used by

Sectional forward Euler converges globally with first-order error Theorem

Cited to discharge well-definedness by Forward Euler for a finite sectional balance with a positivity CFL.

Dependency tree · two levels

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §6.2 (standard reference, not scraped)
Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §§III.2–III.3 (standard reference, not scraped)