{ "scope": "Benchmark of the first-order Ghosh quadrupolar scalar PDE.\n\nDimensionless R=r/mu, Omega=mu*omega, x=cos(theta), u=R*Phi, m=0.\nOn R in [2.5,10], impose homogeneous radial Dirichlet data, regular poles.\nNo horizon/outgoing conditions, QNM claim, time evolution, or all-order metric.\nSource = 16 t^2(1-t)^2 chi_0(x), t=(R-Rin)/(Rout-Rin), chi_0=1/sqrt(2).\nNorm^2 = integral |u|^2 dR dx/f (equivalently |Phi|^2 R^2 dR dx/f).\nThe direct angular collocation uses pointwise multiplication and derivatives,\nnot the Galerkin coupling matrix. Radial collocation is shared and converged.\n\nOriginal implementation for blackhole.org, 2026-09-05. Primary equation:\nGhosh, Franchini, Voelkel, Barausse, Physical Review D 108 (2023) 024038,\narXiv:2303.00088v2, printed pp. 4-5, Eqs. (28)-(34).\nRerun with --output PATH; --verify-saved PATH compares a complete fresh run\nwith every saved field, excluding Python-version provenance, wall-clock time,\nand unresolved last digits. Pinned numerical-library versions still match.\n", "python": "3.9.6", "numpy": "2.0.2", "scipy": "1.13.1", "sympy": "1.14.0", "source_sha256": "e9a6466cd99516d83d6b744f6c47bbc04464451979af718462f6f553684890b7", "annulus": [ 2.5, 10.0 ], "frequencies": [ [ 0.35, -0.025 ], [ 0.55, -0.01 ] ], "epsilons": [ 0.0, 0.25, 0.5, 1.0, 2.0 ], "exact_checks": { "metric_density_series": "1-epsilon*q", "time_density_series": "1-2*epsilon*q", "spatial_density_series": "1", "radial_reduction_residual": "0", "coupling_quadrature_max_errors": [ 6.8486882831564344e-15, 3.524958103184872e-14, 6.439293542825908e-14, 9.903189379656396e-14, 1.366129431801255e-13 ], "maximum_f1": "27/1024 at R=8/3", "maximum_metric_fraction_at_epsilon_2": 0.052734375 }, "manufactured_solution_relative_errors": { "galerkin_R24_lmax6": 2.8903945124243294e-14, "direct_R24_x12": 2.0277341580454847e-14 }, "conventions": { "radial_field": "u=R Phi", "polar_basis": "chi_l=sqrt((2l+1)/2) P_l", "full_sphere_conversion": "Y_l0=chi_l/sqrt(2 pi)", "radial_boundaries": "homogeneous Dirichlet", "poles": "regular; no additional Dirichlet data", "forcing": "16 t^2(1-t)^2 chi_0; t=(R-2.5)/7.5", "empirical_comparison_floor": 1e-12 }, "norm_definition": "squared norm integral |u|^2 dR dx/f, evaluated by Gauss quadrature (100 radial,48 angular); not a physical energy norm", "conditioning_note": "Inverse/condition 1-norm estimates refer only to the unweighted finite nodal matrix and are not continuum resolvent bounds. Response norm ratio tests this one forcing only.", "resolution_floor_note": "Differences below the observed independent-reference convergence floor must not be claimed as resolved continuum errors.", "source_norm": 2.146804632542808, "comparison": [ { "omega_index": 0, "epsilon": 0.0, "reference_radial_degree": 64, "reference_angular_degree": 20, "reference_field_norm": 36.12548271648405, "source_norm": 2.146804632542808, "response_norm_ratio": 16.82755951280709, "inverse_1norm_estimate": 121.94394748700219, "condition_1norm_estimate": 4937445.196808974, "reference_backward_error": 7.9601259696486155e-19, "reference_offgrid_residual_relative": 8.310809473196978e-13, "lmax": 0, "relative_field_error": 7.373237216988205e-14, "omitted_forcing_norm": 0.0, "relative_omitted_forcing": 0.0, "full_offgrid_residual_relative": 6.592184490557234e-13, "full_residual_minus_omitted_relative": 6.592184490557234e-13, "galerkin_backward_error": 2.8994368137774223e-19 }, { "omega_index": 0, "epsilon": 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