centered_2nd_deriv_nonuniform_vertical

Family: finite_difference
Grid family: vertical
Kind: scheme
Accuracy: O(dz_k^2)
Applies to: d2($u), dim=$k
Rule file: discretizations/finite_difference/centered_2nd_deriv_nonuniform_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):

2·((index($u, i + 1) + −1·index($u, i))/(0.5·(index(dz, i) + index(dz, i + 1))) + −1·(index($u, i) + −1·index($u, i + −1))/(0.5·(index(dz, i + −1) + index(dz, i))))/(0.5·(index(dz, i) + index(dz, i + 1)) + 0.5·(index(dz, i + −1) + index(dz, i)))

Source: discretizations/finite_difference/centered_2nd_deriv_nonuniform_vertical.json.