covariant_laplacian_cubed_sphere
Stencil
(ξ, η) coordinates. The
4-point cross stencil carries the orthogonal metric (g^{ξξ}, g^{ηη})
and metric-derivative corrections; the 4 corner cells carry the
cross-metric contribution ±g^{ξη} / (2 h²).Continuous form
Starting from the divergence-form Laplace–Beltrami operator on the gnomonic cubed-sphere panel,
$$\nabla^2 \varphi = \tfrac{1}{J}\bigl[\partial_\xi(J\,g^{\xi\xi}\,\partial_\xi\varphi + J\,g^{\xi\eta}\,\partial_\eta\varphi) + \partial_\eta(J\,g^{\xi\eta}\,\partial_\xi\varphi + J\,g^{\eta\eta}\,\partial_\eta\varphi)\bigr],$$we expand into an orthogonal part plus a cross-metric part:
$$\nabla^2 \varphi = \underbrace{g^{\xi\xi}\,\partial^2_\xi\varphi + g^{\eta\eta}\,\partial^2_\eta\varphi + \tfrac{1}{J}\bigl(\partial_\xi(J g^{\xi\xi})\,\partial_\xi\varphi + \partial_\eta(J g^{\eta\eta})\,\partial_\eta\varphi\bigr)}_{\text{orthogonal}} + \underbrace{2\,g^{\xi\eta}\,\partial^2_{\xi\eta}\varphi + \tfrac{1}{J}\bigl(\partial_\xi(J g^{\xi\eta})\,\partial_\eta\varphi + \partial_\eta(J g^{\xi\eta})\,\partial_\xi\varphi\bigr)}_{\text{cross-metric}}.$$Discrete operator
Substituting centered second-order finite differences for the first and
second derivatives over in-panel cell offsets (Δξ, Δη) ∈ {−1, 0, +1}²
yields a 9-point linear combination of cell values:
with coefficients (uniform isotropic spacing h = dξ = dη = π/(2 N_c)):
offset (Δξ, Δη) | coefficient c_{Δξ,Δη} | contribution |
|---|---|---|
( 0, 0) | −2 (g^{ξξ} + g^{ηη}) / h² | central — orthogonal ∂²_ξ + ∂²_η diagonal |
(+1, 0) | g^{ξξ}/h² + (∂_ξ(J g^{ξξ}) + ∂_η(J g^{ξη})) / (2 h J) | east — orthogonal ∂²_ξ + first-derivative correction |
(−1, 0) | g^{ξξ}/h² − (∂_ξ(J g^{ξξ}) + ∂_η(J g^{ξη})) / (2 h J) | west |
( 0, +1) | g^{ηη}/h² + (∂_η(J g^{ηη}) + ∂_ξ(J g^{ξη})) / (2 h J) | north — orthogonal ∂²_η + first-derivative correction |
( 0, −1) | g^{ηη}/h² − (∂_η(J g^{ηη}) + ∂_ξ(J g^{ξη})) / (2 h J) | south |
(+1, +1) | +g^{ξη} / (2 h²) | NE corner — cross-metric ∂²_{ξη} |
(−1, −1) | +g^{ξη} / (2 h²) | SW corner |
(+1, −1) | −g^{ξη} / (2 h²) | SE corner |
(−1, +1) | −g^{ξη} / (2 h²) | NW corner |
Each metric quantity is resolved at the central cell (i, j) by the
cubed_sphere grid accessor; the rule does not interpolate metric to face
midpoints. The full machine-readable coefficient expressions live in
the JSON rule file.
Metric bindings
| Binding | Resolved by |
|---|---|
J, ginv_xi_xi, ginv_eta_eta, ginv_xi_eta | cubed_sphere accessor at the central cell |
dJgxx_dxi, dJgyy_deta, dJgxe_dxi, dJgxe_deta | metric-derivative tables on the panel |
h | uniform isotropic spacing dξ = dη = π / (2 Nc) |
Cross-panel ghost cells and basis rotation at panel boundaries live in the
cubed_sphere accessor (src/grids/panel_connectivity.jl); the rule
selectors do not carry a panel field — (Δξ, Δη) offsets resolve
across panel seams via the accessor’s connectivity table.
Convergence
Numeric coverage today lives in the canonical Julia test
test/test_laplacian.jl:
the manufactured solution φ(ξ, η) = cos(2ξ)·cos(2η) is evaluated against the
analytic covariant Laplacian (with metric quantities sampled from the gnomonic
metric) at Nc ∈ {8, 16, 32}. The error ratio across each 2× refinement
satisfies e_{k−1}/e_k > 2, consistent with O(h²) asymptotic convergence —
the test asserts that bound directly. The bit-equivalence canonical fixture at
fixtures/canonical/
pins the c24 (Nc = 24) reference output to within 1e-12 relative
tolerance against fv_laplacian from src/operators/laplacian.jl.
(ξ, η) with the gnomonic Jacobian threaded through. Until that
extension lands, the convergence fixture under
[fixtures/convergence/](https://github.com/EarthSciML/EarthSciDiscretizations/blob/main/discretizations/finite_difference/covariant_laplacian_cubed_sphere/fixtures/convergence)
declares applicable: false and the rendered convergence plot is
suppressed. Numeric coverage continues to live at
test/test_laplacian.jl as described above.Reference
- Putman & Lin (2007), JCP 227(1):55–78 — gnomonic cubed-sphere covariant FV operators.
- Imperative reference:
fv_laplacianinsrc/operators/laplacian.jl.