{"id": "diffusion_1d", "title": "Independent 1-D model of the BR2 formulation", "description": "A stand-alone NumPy implementation (it shares only the 1-D basis with the solver) of the same formulation used for viscous terms: strong local gradient, face liftings on Gauss\u2013Lobatto nodes, weak divergence, averaged interface flux with penalty. It solves a diffusion problem for u and a second equation for T that contains the nonlinear work term u \u03c4 of the energy equation, on uniform and non-uniform grids. The checks require the design order p + 1 (within 0.2) for every degree; where they pass, the reduced rates seen for the coupled Navier\u2013Stokes system are not a property of the diffusion discretization alone.", "passed": true, "checks": [{"name": "uniform, p = 1: order of u (diffusion)", "value": 1.9863052530965217, "expected": ">= 1.8", "passed": true}, {"name": "uniform, p = 1: order of T (with work term u \u03c4)", "value": 1.936712431101271, "expected": ">= 1.8", "passed": true}, {"name": "uniform, p = 2: order of u (diffusion)", "value": 3.017960370109446, "expected": ">= 2.8", "passed": true}, {"name": "uniform, p = 2: order of T (with work term u \u03c4)", "value": 3.0913692352691045, "expected": ">= 2.8", "passed": true}, {"name": "uniform, p = 3: order of u (diffusion)", "value": 3.9926849853394453, "expected": ">= 3.8", "passed": true}, {"name": "uniform, p = 3: order of T (with work term u \u03c4)", "value": 4.203432337317868, "expected": ">= 3.8", "passed": true}, {"name": "uniform, p = 4: order of u (diffusion)", "value": 5.003741047973619, "expected": ">= 4.8", "passed": true}, {"name": "uniform, p = 4: order of T (with work term u \u03c4)", "value": 5.461806783378681, "expected": ">= 4.8", "passed": true}, {"name": "non-uniform, p = 1: order of u (diffusion)", "value": 2.267228616664844, "expected": ">= 1.8", "passed": true}, {"name": "non-uniform, p = 1: order of T (with work term u \u03c4)", "value": 2.172997845304726, "expected": ">= 1.8", "passed": true}, {"name": "non-uniform, p = 2: order of u (diffusion)", "value": 3.5915004955901075, "expected": ">= 2.8", "passed": true}, {"name": "non-uniform, p = 2: order of T (with work term u \u03c4)", "value": 3.7599269828748234, "expected": ">= 2.8", "passed": true}, {"name": "non-uniform, p = 3: order of u (diffusion)", "value": 4.555559514817292, "expected": ">= 3.8", "passed": true}, {"name": "non-uniform, p = 3: order of T (with work term u \u03c4)", "value": 4.893350428228524, "expected": ">= 3.8", "passed": true}, {"name": "non-uniform, p = 4: order of u (diffusion)", "value": 5.988806141556627, "expected": ">= 4.8", "passed": true}, {"name": "non-uniform, p = 4: order of T (with work term u \u03c4)", "value": 6.646661279070711, "expected": ">= 4.8", "passed": true}, {"name": "symmetry of the mass-weighted diffusion matrix", "value": 2.5579538487363607e-13, "expected": "< 1e-9", "passed": true}], "charts": [{"id": "error_vs_h", "type": "convergence", "title": "1-D diffusion: error of u", "xlabel": "h", "ylabel": "L2 error", "xscale": "log", "yscale": "log", "series": [{"name": "p = 1: u", "x": [0.125, 0.0625, 0.03125], "y": [0.0010662494243291612, 0.00027240710271070963, 6.87513080523879e-05]}, {"name": "p = 2: u", "x": [0.125, 0.0625, 0.03125], "y": [2.319194693498546e-05, 2.7655358845591516e-06, 3.414150810640796e-07]}, {"name": "p = 3: u", "x": [0.125, 0.0625, 0.03125], "y": [9.01776180815598e-07, 5.717345021204873e-08, 3.5915048503767793e-09]}, {"name": "p = 4: u", "x": [0.125, 0.0625, 0.03125], "y": [6.323046216825358e-09, 1.956129385347722e-10, 6.0970735105129155e-12]}], "reference_slopes": [{"slope": 2, "label": "h^2"}, {"slope": 3, "label": "h^3"}, {"slope": 4, "label": "h^4"}, {"slope": 5, "label": "h^5"}]}], "tables": [{"id": "orders", "title": "Observed orders on the finest pair (\u03b7 = 2)", "columns": ["grid", "p", "L2 error u", "order u", "L2 error T", "order T"], "rows": [["uniform", 1, 6.87513080523879e-05, 1.9863052530965217, 3.6884361977086395e-05, 1.936712431101271], ["uniform", 2, 3.414150810640796e-07, 3.017960370109446, 1.0618399371195919e-07, 3.0913692352691045], ["uniform", 3, 3.5915048503767793e-09, 3.9926849853394453, 5.66699917784589e-10, 4.203432337317868], ["uniform", 4, 6.0970735105129155e-12, 5.003741047973619, 1.3510123607217275e-12, 5.461806783378681], ["non-uniform", 1, 7.894278694571152e-05, 2.267228616664844, 4.12478350482397e-05, 2.172997845304726], ["non-uniform", 2, 4.2636698496937277e-07, 3.5915004955901075, 1.3260978102329737e-07, 3.7599269828748234], ["non-uniform", 3, 4.663825507772114e-09, 4.555559514817292, 7.316089978925358e-10, 4.893350428228524], ["non-uniform", 4, 9.704320159734753e-12, 5.988806141556627, 2.1334192608282478e-12, 6.646661279070711]]}], "provenance": {"version": "0.4.0", "commit": "39979fcf902ad6116eb550b881b4278d4599c8e2", "dirty": false, "describe": "39979fcf90", "device": {"platform": "cpu", "device": "cpu:0", "device_kind": "cpu", "jax": "0.11.2", "x64": true}, "timestamp": "2026-11-12T10:22:38-0500"}, "wall_time": 4.93, "level": "quick"}