divergence_arakawa_c

Family: finite_volume
Grid family: arakawa
Kind: scheme
Accuracy: O(h²)
Applies to: div(F)
Rule file: discretizations/finite_volume/divergence_arakawa_c.json
Tags: #finite-volume #arakawa #c-grid #divergence #staggered

Stencil

Coefficients

selector	stagger	axis	offset	coeff
`arakawa`	`face_x`	`$x`	0	`−1 / dx`
`arakawa`	`face_x`	`$x`	+1	`+1 / dx`
`arakawa`	`face_y`	`$y`	0	`−1 / dy`
`arakawa`	`face_y`	`$y`	+1	`+1 / dy`

The rule declares an arakawa_stagger enum mapping stagger names (cell_center, face_x, face_y, vertex) to integer codes — the runtime uses these to dispatch location_centers(grid, ...) per stagger.

requires_locations: ["face_x", "face_y"], emits_location: "cell_center".

Discrete operator

Let cell $(i, j)$ have center $(x_i, y_j)$ and uniform spacings $\Delta x, \Delta y$. The flux components live on the staggered C-grid: $F_x$ at the east/west faces $(x_{i\pm 1/2}, y_j)$ and $F_y$ at the north/south faces $(x_i, y_{j\pm 1/2})$. With the four faces of cell $(i, j)$ indexed by their stagger offset $0 / +1$ along the relevant axis, the rule emits the cell-centered divergence

$$\bigl(\nabla \!\cdot\! F\bigr)_{i,j} \;\approx\; \frac{F_x^{\,i+1/2,\,j} - F_x^{\,i-1/2,\,j}}{\Delta x} \;+\; \frac{F_y^{\,i,\,j+1/2} - F_y^{\,i,\,j-1/2}}{\Delta y}.$$

Each one-dimensional difference is the centered face-to-center stencil applied to a face-staggered field. Taylor-expanding $F_x$ about the cell center,

$$F_x^{\,i\pm 1/2,\,j} \;=\; F_x(x_i, y_j) \;\pm\; \tfrac{\Delta x}{2}\,\partial_x F_x \;+\; \tfrac{\Delta x^{2}}{8}\,\partial_x^{2} F_x \;\pm\; \tfrac{\Delta x^{3}}{48}\,\partial_x^{3} F_x \;+\; O(\Delta x^{4}),$$

so subtracting the two faces and dividing by $\Delta x$ cancels the even-order terms exactly:

$$\frac{F_x^{\,i+1/2,\,j} - F_x^{\,i-1/2,\,j}}{\Delta x} \;=\; \partial_x F_x \;+\; \tfrac{\Delta x^{2}}{24}\,\partial_x^{3} F_x \;+\; O(\Delta x^{4}).$$

The same expansion in $y$ gives $\partial_y F_y + (\Delta y^{2}/24)\,\partial_y^{3} F_y + O(\Delta y^{4})$. Summing the two contributions yields a second-order accurate divergence with a purely dispersive leading error $\tfrac{\Delta x^{2}}{24}\,\partial_x^{3} F_x + \tfrac{\Delta y^{2}}{24}\,\partial_y^{3} F_y$. Because $F_x$ and $F_y$ are sampled at the very faces whose normal flux they represent, no interpolation is needed — the C-grid stagger makes the discrete divergence mimetic with respect to the analytic Gauss identity $\int_{V_{ij}} \nabla\!\cdot\!F\,dV = \oint_{\partial V_{ij}} F\!\cdot\!\hat{n}\,dS$, and exact for flux fields that are piecewise-linear in each direction.

Convergence

Convergence plot pending fixture activation. The empirical convergence plot will be generated once the fixture under [fixtures/convergence/](https://github.com/EarthSciML/EarthSciDiscretizations/blob/main/discretizations/finite_volume/divergence_arakawa_c/fixtures/convergence) flips from applicable: false to expected_min_order = 2. The arakawa-staggered manufactured-solution dispatch needed by Layer B (apply_stencil_2d_arakawa in EarthSciSerialization.jl/src/mms_evaluator.jl) has landed; the fixture flip itself is the remaining gate.