centered_2nd_uniform_vertical
Stencil
Coefficients
| selector kind | axis | stagger | offset | coeff |
|---|---|---|---|---|
vertical | $k | face_bottom | 0 | −1 / h |
vertical | $k | face_top | 0 | +1 / h |
Combined as +. The vertical selector kind addresses face-staggered
samples of u: at cell i, face_bottom with offset 0 reads the face
at the bottom of the cell, and face_top with offset 0 reads the face at
the top of the cell. The vertical accessor exposes the level spacing h
directly so the rule does not need to know whether the column is in z,
p, or σ coordinates.
Discrete operator
For face samples $u_{i+1/2}$ on a uniform vertical axis with cell spacing $h$, the rule produces a cell-centered derivative at cell $i$ (located at $z_i$, the midpoint of the cell’s two faces $z_{i-1/2}$ and $z_{i+1/2}$):
$$\bigl(\partial_k u\bigr)_i \;\approx\; \frac{u_{i+1/2} - u_{i-1/2}}{h}.$$Taylor-expanding $u_{i\pm 1/2}$ about cell center $z_i$,
$$\frac{u_{i+1/2} - u_{i-1/2}}{h} \;=\; u'(z_i) \;+\; \frac{h^{2}}{24}\,u'''(z_i) \;+\; \mathcal{O}(h^{4}),$$so the leading truncation error is $\tfrac{h^{2}}{24},u’’’(z_i)$, giving the advertised $\mathcal{O}(h^{2})$ accuracy. The expression is anti-symmetric in the two adjacent faces, so the operator is non-dissipative — error appears as dispersion only, with no built-in numerical diffusion.
The vertical accessor binds h to whichever coordinate the column actually
uses ($\Delta z$, $\Delta p$, or $\Delta\sigma$); the discrete operator
above is unchanged. Non-uniform spacing is out of scope for this rule —
the uniform axis is enforced by the vertical selector kind.
Convergence
u(z) = sin(2πz), column [0, 1],
face-staggered samples → cell-centered derivative. Slope matches the −2 reference.Fixture: discretizations/finite_difference/centered_2nd_uniform_vertical/fixtures/convergence/