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Periodic PIC deposition and potential gather are adjoint

Statement

Let $ρ_{j}^{+} = h^{- d} \sum_{p} q_{p} w_{j} (X_{p})$ be the deposited density before background subtraction for an admissible algebraic shape $w_{j}$ of Periodic PIC deposition shape and gather operators, and let $G_{X}$ be its scalar potential gather. For every grid scalar $ϕ$ , $⟨ ρ^{+}, ϕ ⟩_{h} = \sum_{p = 1}^{P} q_{p} (G_{X} ϕ)_{p} .$ For the neutralized density $ρ = ρ^{+} - Q / L^{d}$ the corresponding identity is $⟨ ρ, ϕ ⟩_{h} = \sum_{p} q_{p} (G_{X} ϕ)_{p} - Q \overset{ˉ}{ϕ} .$ Thus the background term vanishes in the zero-mean potential gauge.

Facts & Assumptions

Given: The grid inner product, particle positions and charges, and the deposition and gather formulas.

[F1]

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

[F2]

Periodic PIC deposition shape and gather operators: $ρ_{j}^{+} = h^{- d} \sum_{p} q_{p} w_{j} (X_{p})$ and $(G_{X} ϕ)_{p} = \sum_{j} w_{j} (X_{p}) ϕ_{j}$ .

Proof

1.1givenF1F2algebra

Substitute [F2] into the left pairing and cancel $h^{d} h^{- d}$ : $⟨ ρ^{+}, ϕ ⟩_{h} = h^{d} \sum_{j} h^{- d} \sum_{p} q_{p} w_{j} (X_{p}) ϕ_{j} = \sum_{p} q_{p} \sum_{j} w_{j} (X_{p}) ϕ_{j} .$ The finite inner sum is $(G_{X} ϕ)_{p}$ by [F2], proving the first identity.

2.1F1step 1.1algebra∎

The background contributes ${⟨ Q / L^{d}, ϕ ⟩}_{h} = Q L^{- d} h^{d} \sum_{j} ϕ_{j} = Q \overset{ˉ}{ϕ}$ by [F1]. Subtract this term from step 1.1 to obtain the second identity.

Depends on

Used by

One-dimensional periodic CIC deposition and adjoint potential gather Example

Dependency tree · one level

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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)