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Periodic PIC deposition conserves charge and satisfies Poisson compatibility

Statement

For positions $X_{1}, \dots, X_{P} \in T_{L}^{d}$ and real charges $q_{1}, \dots, q_{P}$ , define $Q = \sum_{p} q_{p}$ and an admissible algebraic assignment shape $w_{j}$ as in Periodic PIC deposition shape and gather operators. Then $h^{d} \sum_{j \in J_{N}} h^{- d} \sum_{p = 1}^{P} q_{p} w_{j} (X_{p}) = Q .$ Consequently the neutralized source $ρ_{j} (X) = h^{- d} \sum_{p} q_{p} w_{j} (X_{p}) - Q / L^{d}$ has mean zero and is in the range of the periodic Poisson operator; it therefore has a unique mean-zero potential for the gauge $\overset{ˉ}{ϕ} = 0$ by Periodic centered-difference Poisson solvability and zero-mean gauge.

Facts & Assumptions

Given: The finite periodic grid, particle positions and charges, the normalized assignment shape, and the periodic Poisson operator.

[F1]

Periodic PIC deposition shape and gather operators: $ρ_{j} = h^{- d} \sum_{p} q_{p} w_{j} (X_{p}) - Q / L^{d}$ , and an algebraic assignment has nonnegative weights.

[F2]

Periodic PIC deposition shape and gather operators: every algebraic assignment shape satisfies $\sum_{j} w_{j} (x) = 1$ for every particle location.

[F3]

Periodic centered-difference Poisson solvability and zero-mean gauge: the range of $A_{h}$ is exactly the zero-mean grid subspace, and every zero-mean source has a unique mean-zero solution.

Proof

1.1givenF1F2algebra

Before subtracting the background, sum the assigned density with cell volume. Since the sum is finite, interchange its particle and grid sums; [F2] gives $h^{d} \sum_{j} h^{- d} \sum_{p} q_{p} w_{j} (X_{p}) = \sum_{p} q_{p} \sum_{j} w_{j} (X_{p}) = \sum_{p} q_{p} = Q .$

2.1F1step 1.1algebra

The background has total grid charge $h^{d} ∣ J_{N} ∣ Q / L^{d} = L^{d} Q / L^{d} = Q$ because $h^{d} ∣ J_{N} ∣ = L^{d}$ . Hence $⟨ ρ, 1 ⟩_{h} = 0$ , equivalently $\overset{ˉ}{ρ} = 0$ .

3.1F3step 2.1∎

By [F3], this zero-mean source has a solution and fixing its mean to zero selects exactly one potential.

Depends on

Used by

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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)
A. Myers, P. Colella, and B. Van Straalen, A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov–Poisson Equation, Section 2.2 (standard reference, not scraped)