spherical_laplacian_nonuniform

Family: finite_difference
Grid family: cartesian
Kind: scheme
Accuracy: O(dr_i^2)
Applies to: spherical_laplacian($u), dim=$r
Rule file: discretizations/finite_difference/spherical_laplacian_nonuniform.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 spherical_laplacian on the pattern spherical_laplacian($u), dim=$r for grid family cartesian.

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):

out[$r] = 2·((0.5·(index(r, $r) + index(r, $r + 1)))²·(index($u, $r + 1) − index($u, $r))/index(dr, $r + 1) − (0.5·(index(r, $r − 1) + index(r, $r)))²·(index($u, $r) − index($u, $r − 1))/index(dr, $r))/(index(r, $r)²·(index(dr, $r) + index(dr, $r + 1)))

Source: discretizations/finite_difference/spherical_laplacian_nonuniform.json.