centered_2nd_deriv_uniform_vertical

Family: finite_difference
Grid family: vertical
Kind: scheme
Accuracy: O(dz^2)
Applies to: d2($u), dim=$k
Rule file: discretizations/finite_difference/centered_2nd_deriv_uniform_vertical.json

This page is auto-generated from the rule's JSON by tools/render_doc_plots.py. The discrete operator below is pretty-printed directly from the rule's replacement AST using EarthSciSerialization (earthsci_toolkit). Rules with a hand-authored page carry richer prose plus stencil and convergence diagrams; see the curated entries linked from the rule index.

Pattern

Matches the §4.2 operator d2 on the pattern d2($u), dim=$k for grid family vertical.

Discrete operator

Pretty-printed from the rule’s replacement AST ($-prefixed names are pattern variables bound by applies_to; grid metrics such as dx/dz are resolved by the grid):

1/(dz·dz)·index($u, i + 1) + −2/(dz·dz)·index($u, i)

Source: discretizations/finite_difference/centered_2nd_deriv_uniform_vertical.json.