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One-dimensional periodic CIC deposition and adjoint potential gather

Statement

Take $L = 1$ , $N = 5$ , $h = 1 / 5$ , and one particle of charge $q = 2$ at $X = h / 3$ . Define the one-dimensional cloud-in-cell (CIC) weights by $w_{j} (x) = \max {1 - d_{L} (x, x_{j}) / h, 0}$ , where $d_{L} (x, x_{j}) = \min_{k \in Z} ∣ x - x_{j} + k L ∣$ is the periodic distance. The only nonzero weights at $X$ are $w_{0} (X) = 2 / 3$ and $w_{1} (X) = 1 / 3$ . The deposited density before neutralization is $(20 / 3, 10 / 3, 0, 0, 0)$ , and subtracting the uniform background $Q / L = 2$ gives $ρ = (14 / 3, 4 / 3, - 2, - 2, - 2), h \sum_{j = 0}^{4} ρ_{j} = 0.$ For every grid potential $ϕ$ , the adjoint identity is $⟨ ρ, ϕ ⟩_{h} = 2 (\frac{2}{3} ϕ_{0} + \frac{1}{3} ϕ_{1}) - 2 \overset{ˉ}{ϕ} .$

The diagram shows the local CIC stencil between nodes $0$ and $1$ ; indices on the full grid are understood modulo $5$ .

\begin{tikzpicture}[x=2cm,y=1cm,>=stealth,every node/.style={font=\small}] \draw[->] (-0.35,0) -- (2.35,0) node[right] { $x$ }; \foreach \j in {0,1,2} { \draw (\j,0.07) -- (\j,-0.07); \fill (\j,0) circle (1.4pt); \node[below=3pt] at (\j,-0.07) {node \j}; } \fill[red] (0.333,0.55) circle (1.6pt); \node[above] at (0.333,0.55) { $X = h / 3$ }; \draw[dashed,red] (0.333,0.52) -- (0,0.04); \draw[dashed,red] (0.333,0.52) -- (1,0.04); \node[red,left] at (-0.05,0.45) { $w_{0} = 2 / 3$ }; \node[red,right] at (0.95,0.45) { $w_{1} = 1 / 3$ }; \draw[<->] (0,0.26) -- node[above] { $h$ } (1,0.26); \end{tikzpicture}

Facts & Assumptions

Given: The five-node periodic grid, a particle of charge $2$ at $h / 3$ , the CIC weights, and the assignment/gather conventions.

[F1]

Periodic uniform PIC grid and inner product: $⟨ u, v ⟩_{h} = h \sum_{j} u_{j} v_{j}$ and $\overset{ˉ}{v} = L^{- 1} ⟨ v, 1 ⟩_{h}$ .

[F2]

Periodic PIC deposition shape and gather operators: $ρ_{j}^{+} = h^{- 1} q w_{j} (X)$ , $ρ_{j} = ρ_{j}^{+} - Q / L$ , and $(G_{X} ϕ) = \sum_{j} w_{j} (X) ϕ_{j}$ .

[F3]

Periodic PIC deposition conserves charge and satisfies Poisson compatibility: normalized assignment conserves total charge and the neutralized source has zero mean.

[F4]

Periodic PIC deposition and potential gather are adjoint: $⟨ ρ, ϕ ⟩_{h} = \sum_{p} q_{p} (G_{X} ϕ)_{p} - Q \overset{ˉ}{ϕ}$ .

[F5]

Periodic PIC deposition shape and gather operators: an algebraic assignment shape need only be nonnegative and sum to one; smoothness is an additional condition for the smooth fixed- $ε$ branch.

Proof

1.1F1F2givenalgebra

Since $h^{- 1} = 5$ , the two nonzero assigned densities are $5 \cdot 2 \cdot (2 / 3) = 20 / 3$ and $5 \cdot 2 \cdot (1 / 3) = 10 / 3$ . The background is $Q / L = 2$ , so the neutralized values are $14 / 3, 4 / 3, - 2, - 2, - 2$ ; their sum is $6 - 6 = 0$ , hence their cell-weighted sum is zero.

1.2F2givenalgebra

For any point in a periodic cell $[x_{j}, x_{j + 1}]$ , with its right endpoint interpreted modulo $L$ , the CIC formula gives $w_{j} = (x_{j + 1} - x) / h$ and $w_{j + 1} = (x - x_{j}) / h$ ; all other weights vanish. Both are nonnegative and sum to one. At a grid node this reduces to the single weight one. Hence the stated CIC family is a nonnegative normalized assignment shape.

2.1F1F2F3F4step 1.1step 1.2algebra

The pre-neutralized cell-weighted sum is $h (20 / 3 + 10 / 3) = 2$ , verifying charge conservation directly; [F3] also applies because step 1.2 proves the shape partition. By [F4], the potential pairing equals $2 (\frac{2}{3} ϕ_{0} + \frac{1}{3} ϕ_{1}) - 2 \overset{ˉ}{ϕ}$ . Since $\overset{ˉ}{ϕ} = \frac{1}{5} \sum_{j = 0}^{4} ϕ_{j}$ , the formula includes exactly the contribution of the uniform background.

3.1F5algebra∎

Near node $x_{0} = 0$ , the periodic distance formula gives $w_{0} (x) = 1 + x / h$ for $- h < x < 0$ and $w_{0} (x) = 1 - x / h$ for $0 < x < h$ . The one-sided derivatives at the node are $1 / h$ and $- 1 / h$ , so the CIC weight is continuous but not differentiable at this grid crossing. Therefore CIC regularity alone does not supply the $C^{4}$ vector-field hypothesis of the RK4 error theorem; the page's time-error claim uses the separate smooth fixed- $ε$ branch.

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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)