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The periodic PIC Poisson algorithm returns the unique gauged solution

Statement

For every real zero-mean source $s$ on the grid of Periodic uniform PIC grid and inner product, Fourier diagonal solver for the periodic centered-difference Poisson equation terminates and returns the unique $ϕ$ satisfying $A_{h} ϕ = s$ and $\overset{ˉ}{ϕ} = 0$ . Its incompatible-source branch is taken exactly when the equation has no solution.

Facts & Assumptions

Given: A real grid source and the exact-arithmetic finite Fourier procedure.

[F1]

Periodic centered-difference Poisson solvability and zero-mean gauge: the Fourier characters diagonalize $A_{h}$ , $λ_{h} (k) > 0$ for $k \neq 0$ , and a zero-mean source has a unique zero-mean solution.

[F2]

Fourier diagonal solver for the periodic centered-difference Poisson equation: the procedure checks $\overset{ˉ}{s}$ , sets ${\hat{ϕ}}_{0} = 0$ , divides each other coefficient by $λ_{h} (k)$ , and applies the inverse finite Fourier transform.

Proof

1.1F1F2algebra

The source and grid contain finitely many real numbers, so computing its mean and finite Fourier transform terminates. If $\overset{ˉ}{s} \neq 0$ , then the constant Fourier coefficient of $s$ is nonzero while $A_{h}$ annihilates constants and has zero constant coefficient on every output, so $A_{h} ϕ = s$ is impossible.

2.1F1F2step 1.1algebra

If $\overset{ˉ}{s} = 0$ , its zero Fourier coefficient is zero. By [F1], every other eigenvalue is positive, so all divisions in [F2] are defined. For real input, ${\hat{s}}_{- k} = \overline{{\hat{s}}_{k}}$ and $λ_{h} (- k) = λ_{h} (k)$ ; therefore the returned coefficients have the same conjugate symmetry, and pairing the $k$ and $- k$ terms in the inverse transform shows every $ϕ_{j}$ is real. For every nonzero mode, the returned coefficient satisfies $λ_{h} (k) {\hat{ϕ}}_{k} = {\hat{s}}_{k}$ , and the zero mode satisfies $0 = {\hat{s}}_{0}$ . Inverse Fourier transformation therefore gives $A_{h} ϕ = s$ and ${\hat{ϕ}}_{0} = 0$ gives $\overset{ˉ}{ϕ} = 0$ .

3.1F1step 2.1∎

The coefficient list is finite, hence inverse transformation terminates. Uniqueness of the returned gauged solution follows from [F1].

Depends on

Used by

Classical RK4 step for the fixed-grid regularized PIC particle ODE Algorithm

Dependency tree · one level

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Sources

Benjamin Seibold, MIT 18.336 Lecture 5, Finite Difference Approximation (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)