Divergence

SPH divergence of a field. Kernel: coreOperations/wp_divergence.py (computeSPHDivergenceTensor_Kernel); JVP coreOperations/wp_divergenceJVP.py.

Equation

$$\nabla\!\cdot\! \mathbf{f}_i \;=\; \sum_{j \in \mathcal{N}(i)} \mathrm{divergenceProduct}(\mathrm{coeff}_{ij},\, \nabla W_{ij}),$$

where $\mathrm{coeff}_{ij}$ is the same coefficient table as Gradient (Naive / Symmetric / Difference / Summation, with the same apparent-volume $V_j$ and grad-h rescaling), and $\nabla W_{ij}$ is the same CRK-then-renormalized kernel gradient. Only the combination with the gradient differs:

• standard mode (default): for a field of shape [N, ..., d] the last axis is the spatial one and the contribution is the component sum $\sum_{\alpha=1}^{d} \mathrm{coeff}_{ij}[\alpha]\; \partial_\alpha W_{ij}$, contracted over the spatial axis — a vector field [N, d] yields a scalar output [N];
• dot mode (divergenceDotMode=True): the field is a matrix [N, d, d] and the contraction is over the last index only, $\sum_\alpha \mathrm{coeff}_{ij}[\beta, \alpha]\; \partial_\alpha W_{ij}$ for each remaining index $\beta$ — a tensor field yields [N, d].

Output shape: inputShape[:-1] in standard mode, inputShape[1:] in dot mode (see _computeSPHDivergence_stateBackend).

Public API

warpOperation(
    queryParticles, operationProperties, domain,
    queryValues=f, referenceValues=f,      # required
    consistentDivergence=False,
    queryVolumes=None, referenceVolumes=None,
    crkState=None, gradHState=None, renormalizationState=None,
    adjacency=None,
)

with operation == WarpOperation.Divergence.

Option	Default	Effect
gradientMode	Naive	coefficient scheme (same table as Gradient)
divergenceDotMode	False	matrix-field contraction over the last index (output keeps one axis)
consistentDivergence	False	scales the apparent volume by the density ratio: $V_j \leftarrow V_j\,\rho_j/\rho_i$ (or $V_j \leftarrow m_j/\rho_i$ without supplied volumes) — the "consistent" form of $\nabla\!\cdot\!(\rho^{-1}\nabla f)$-type bookkeeping
supportMode / operationMode / corrections / calibration	—	as in Gradient

JVP / HVP

Value JVP — same relaunch-on-tangents identity as every value-having operator (tests/operations/test_forward_mode_value_jvp.py).

Geometry JVP — the product rule with vector-valued coefficients:

$$d(\nabla\!\cdot\!\mathbf{f})_i \;=\; \sum_j \Big[ \mathrm{dcoeff}_{ij}\cdot G_{ij} + \mathrm{coeff}_{ij}\cdot dG_{ij} \Big] \;+\; \sum_j (df_i A_{ij} + df_j B_{ij})\cdot G_{ij}\;\text{(fused)},$$

with the same $A/B/dA/dB$ weights as Gradient (_jvpCommon.gradientWeightsJVP) and $G/dG$ from computeKernelGradientCRKJVP followed by the renormalization product rule $dG = dL_i G + L_i\, dG$.

Scope restrictions (enforced centrally in operations.py):

• divergenceDotMode and consistentDivergence are not in the derived math — the geometry JVP rejects both.
• CRK / renormalization tangents: same rules as Gradient (including simultaneous use); gradHState unsupported; query-mass tangents rejected.

Gated by tests/operations/test_forward_mode_geometry_jvp_divergence.py.

No HVP exists for Divergence.

Tests

• tests/operations/test_operations_core.py — shape/finiteness.
• tests/operations/test_operations_consistency.py — trace of the Difference-scheme Gradient equals the Difference-scheme Divergence (interior mask), the key cross-operator identity.
• tests/operations/test_forward_mode_value_jvp.py, tests/operations/test_forward_mode_geometry_jvp_divergence.py, tests/operations/test_grid_modes.py.

See also

Gradient · Curl · Laplacian