{"id": "density_wave", "title": "Advected density wave", "description": "A smooth density wave carried by a uniform flow is an exact solution of the Euler equations. It isolates the convective operator and the explicit time integrator. The density is advected linearly, so the design order p + 1 of the split-form DGSEM with a Roe interface flux is what the checks require.", "passed": true, "checks": [{"name": "p = 1: observed order (finest pair)", "value": 1.9262429478281593, "expected": ">= 1.80 (design rate p + 1 = 2, tolerance 0.2)", "passed": true}, {"name": "p = 2: observed order (finest pair)", "value": 2.9942544457719587, "expected": ">= 2.80 (design rate p + 1 = 3, tolerance 0.2)", "passed": true}, {"name": "p = 3: observed order (finest pair)", "value": 3.9975254331601597, "expected": ">= 3.80 (design rate p + 1 = 4, tolerance 0.2)", "passed": true}, {"name": "p = 4: observed order (finest pair)", "value": 5.011280990433305, "expected": ">= 4.80 (design rate p + 1 = 5, tolerance 0.2)", "passed": true}], "charts": [{"id": "error_vs_h", "type": "convergence", "title": "Density error after convection", "xlabel": "h", "ylabel": "L2 error of density", "xscale": "log", "yscale": "log", "series": [{"name": "p = 1", "x": [0.125, 0.0625, 0.03125], "y": [0.024451122777660067, 0.007168468513777493, 0.0018861206754589121]}, {"name": "p = 2", "x": [0.125, 0.0625, 0.03125], "y": [0.0012552451438290243, 0.0001620145064347416, 2.0332627263389384e-05]}, {"name": "p = 3", "x": [0.125, 0.0625, 0.03125], "y": [5.2276133502289416e-05, 3.4487335522713583e-06, 2.1591587692026288e-07]}, {"name": "p = 4", "x": [0.125, 0.0625, 0.03125], "y": [2.07359505680332e-06, 6.88057760706644e-08, 2.133432972397772e-09]}], "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": "Errors and observed orders", "columns": ["p", "N", "L2 error", "order"], "rows": [[1, 8, 0.024451122777660067, null], [1, 16, 0.007168468513777493, 1.7701638775480073], [1, 32, 0.0018861206754589121, 1.9262429478281593], [2, 8, 0.0012552451438290243, null], [2, 16, 0.0001620145064347416, 2.953774243802353], [2, 32, 2.0332627263389384e-05, 2.9942544457719587], [3, 8, 5.2276133502289416e-05, null], [3, 16, 3.4487335522713583e-06, 3.9220138628854024], [3, 32, 2.1591587692026288e-07, 3.9975254331601597], [4, 8, 2.07359505680332e-06, null], [4, 16, 6.88057760706644e-08, 4.913460693299226], [4, 32, 2.133432972397772e-09, 5.011280990433305]]}], "provenance": {"version": "0.4.0", "commit": "39979fcf902ad6116eb550b881b4278d4599c8e2", "dirty": false, "describe": "39979fcf90", "device": {"platform": "gpu", "device": "cuda:0", "device_kind": "NVIDIA GeForce RTX 3060 Ti", "jax": "0.11.2", "x64": true}, "timestamp": "2026-11-12T10:22:56-0500"}, "wall_time": 40.98, "notes": "Observed order minus the design rate p + 1 on the finest mesh pair, over all degrees: -0.07 to +0.01.", "level": "quick"}