1. Grid -> Particle Interpolation

For a particle inside a host cell with local coordinates \((a_1,a_2,a_3)\in[0,1]^3\), trilinear interpolation uses 8 source-point weights:

\[ \phi_p = \sum_{m=1}^{8} w_m(a_1,a_2,a_3)\,\phi_m, \qquad \sum_m w_m = 1. \]

PICurv supports two grid-to-particle interpolation methods, selectable at runtime via the interpolation_method YAML key (or -interpolation_method flag).

1.1 Trilinear (Direct Cell-Center) — Default

The recommended method. For each particle, the 8 nearest cell centers are identified via octant detection (based on existing host-cell weights), forming a "dual cell." Trilinear interpolation is performed directly from these cell-centered field values to the particle position.

Key properties:

Second-order accurate on both uniform and curvilinear grids.
No intermediate staging: operates directly on the ghosted cell-centered field (lUcat, lDiffusivity, etc.) and ghosted cell center coordinates (lCent). No extra ghost exchange is needed beyond what the solver already provides.
Boundary handling: at non-periodic boundaries, the dual-cell octant is clamped to use only physical cell centers, and trilinear weights are left unclamped (may exceed [0,1]) to provide second-order linear extrapolation. Weights still sum to 1.0.
Periodic boundaries: PETSc supplies wrapped field indices, while ApplyPeriodicCorrectionsToCellCentersAndSpacing() supplies translated coordinate images. This interpolation path is Eulerian-only; particles are not currently wrapped across periodic boundaries.

Code path: InterpolateEulerFieldToSwarm dispatches to InterpolateEulerFieldFromCenterToSwarm.

1.2 Corner-Averaged (Legacy)

The original two-stage method:

Center -> Corner: average surrounding cell-center values to each grid node via InterpolateFieldFromCenterToCorner (unweighted sum/count).
Corner -> Particle: standard trilinear interpolation from the 8 corner values of the host cell.

This is second-order on uniform Cartesian grids but degrades to first-order on curvilinear grids because the unweighted averaging does not account for the asymmetric placement of cell centers around nodes.

Code path: InterpolateEulerFieldToSwarm dispatches to InterpolateEulerFieldFromCornerToSwarm.

1.3 Shared Kernels

Both methods share the same underlying trilinear kernels:

scalar kernel: TrilinearInterpolation_Scalar
vector kernel: TrilinearInterpolation_Vector
weight computation: ComputeTrilinearWeights
field-by-field dispatcher: InterpolateEulerFieldToSwarm
stage wrapper: InterpolateAllFieldsToSwarm

1.4 Configuration

YAML (solver.yml):

interpolation:

method: "Trilinear" # default; or "CornerAveraged"

C flag: -interpolation_method 0 (Trilinear) or -interpolation_method 1 (CornerAveraged).

Enum: InterpolationMethod in include/variables.h. Stored in SimCtx.interpolationMethod.

2. Particle -> Grid Scatter and Normalization

Scatter computes per-cell sums, then normalizes by particle count:

\[ \bar{\phi}_{i,j,k} = \begin{cases} \dfrac{\sum_{p\in cell(i,j,k)} \phi_p}{N_{i,j,k}}, & N_{i,j,k}>0, \\ 0, & N_{i,j,k}=0. \end{cases} \]

Code touchpoints:

particle count accumulation: CalculateParticleCountPerCell
scatter orchestrator: ScatterAllParticleFieldsToEulerFields

Current standard scatter path actively maps particle Psi to Eulerian Psi; additional fields are scaffolded and can be enabled with matching vector/DM contracts.

3. Accuracy and Stability Considerations

Coupling quality depends on:

accurate host-cell IDs and interpolation weights,
ghost synchronization before interpolation/scatter,
consistent field DOF/DM association,
avoiding stale particle-count vectors between scatter calls.

Inconsistency in any of these usually appears as noisy particle statistics or nonphysical reconstructed Eulerian fields.

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