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Kernels & Green's function

The PIC solver has two orthogonal choices: the particle-mesh shape function (how charge is spread to the grid and field gathered back) and the Green's function (how Poisson is solved).

Particle-mesh shape: CIC vs TSC

CIC TSC
Order 1st (linear B-spline) 2nd (quadratic B-spline)
Cells per particle 8 (one corner each octant) 27 (3×3×3 around centre)
Cost ~1× ~3.4×
Default yes (legacy) recommended for noise-sensitive cases

When CIC is fine

For lattices with frequent RF cavities (HWR, SSR1, SSR2, LB650, HB650 in PIP-II), CIC is sufficient — the RF restoring force damps grid noise. The 3% PIC-kernel calibration offset documented in HELIX's internal memory notes is separate from grid-noise issues — it's a known calibration bias (see PIP-II validation).

When TSC is needed

For long no-RF transports (BTL — beam transport lines), grid noise + self-force aliasing in CIC accumulate without RF damping and ε_z grows unphysically. Switching to TSC drops the artificial growth rate substantially:

BTL stage CIC ε_z growth TSC ε_z growth TraceWin reference
MEBT exit → BTL end ×14 ×1.6 ×1.7

TSC is the recommended kernel for BTL-style sections.

TSC grid convention

The TSC weights anchor the deposited charge's centroid exactly half a cell low relative to the particle position (verified 2026-07-25). Deposit and gather share the convention, so it cancels in every space-charge kick — no beam-dynamics number is affected. Only raw density maps read directly off the ρ grid are shifted by −dx/2. Any future change must move deposit and gather together.

Performance

TSC is ~3.4× more expensive per step than CIC. For PIP-II's full linac with frequent RF, this is unnecessary cost. For BTL-only or DC-only runs, TSC is essential.

Green's function: IGF vs point

IGF point
Source Qiang et al. PRSTAB 9, 044204 (2006) sampled 1/(4πε₀ r)
Symmetry symmetric, no near-source bias sampled, has near-source artefacts
Default yes legacy reproduction

The Integrated Green Function (IGF) is the closed-form 8-point cell-difference of the analytic Coulomb antiderivative — it's the exact Green's function for a uniform-density unit cube. The "point" Green's function samples 1/(4πε₀ r) at cell centres with a self-potential regulariser at r=0; it has been shown to introduce grid-resolution bias near the source.

Always use IGF. The "point" option is kept only for legacy regression reproduction.

Configuration

from linac_gen.core.config import SpaceChargeConfig

# PIP-II-style linac with RF cavities (nx=ny=nz default to 96):
sc_default = SpaceChargeConfig(
    kernel="cic",           # fine with frequent RF
    green_kind="igf",
)

# For BTL or other long no-RF transport:
sc_btl = SpaceChargeConfig(
    kernel="tsc",           # required to avoid ε_z runaway
    green_kind="igf",
)

Cross-references

← PIC solver · Continue to DC mode →