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Smooth periodic PIC assignment weights have bounded derivatives

Statement

For fixed $ε > 0$ and periodic grid, the smooth-branch weights $w_{j}^{ε, h}$ of Periodic PIC deposition shape and gather operators are $C^{\infty}$ on $T_{L}^{d}$ . For every multi-index $β$ , $\max_{j} ∥ \partial^{β} w_{j}^{ε, h} ∥_{\infty} < \infty$ . They are nonnegative and sum to one at every point.

Facts & Assumptions

Given: The explicit smooth periodic kernel $η_{ε}$ , its sampled normalizer $Z_{ε, h}$ , and the displayed weights.

[F1]

Periodic PIC deposition shape and gather operators: $Z_{ε, h} (x) = h^{d} \sum_{j} η_{ε} (x_{j} - x)$ and $w_{j}^{ε, h} (x) = h^{d} η_{ε} (x_{j} - x) / Z_{ε, h} (x)$ , where $Z_{ε, h} (x) > 0$ .

[F2]

Periodic PIC regularization gives a smooth compatible Poisson solution: $η_{ε}$ is strictly positive and $C^{\infty}$ on the compact torus.

Proof

1.1F1F2algebra

By [F2], each numerator in [F1] is a smooth periodic function. The finite sum defining $Z_{ε, h}$ is smooth and strictly positive; compactness gives a positive minimum. The reciprocal of a positive smooth function is smooth, so every ratio $w_{j}^{ε, h}$ is $C^{\infty}$ . Each derivative is continuous on the compact torus and hence bounded; there are finitely many indices $j$ , so their maximum is finite.

2.1F1step 1.1algebra∎

Each weight is nonnegative by [F1] and positivity of $η_{ε}$ . Summing [F1] over $j$ gives $\sum_{j} w_{j}^{ε, h} (x) = Z_{ε, h} (x) / Z_{ε, h} (x) = 1$ .

Depends on

Used by

Dependency tree · two levels

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Sources

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Sections 2-2-3 and 5-2 (standard reference, not scraped)