{"id": "mms_euler", "title": "Manufactured solution: Euler", "description": "A smooth trigonometric state is forced to be an exact steady solution by a source term generated with automatic differentiation of the physics kernels. Dirichlet data enter weakly through the interface Riemann flux. The nonlinear system is solved with the pseudo-transient Newton\u2013Krylov solver, started from the nodal values of the exact solution, whose residual is the truncation error. The study therefore verifies the spatial operator and the boundary treatment, and that the implicit solver removes that residual; it does not test convergence from a distant initial state (see the solver-convergence study). Curved meshes are warped by 5 % with degree-4 geometry.", "passed": true, "checks": [{"name": "p = 1, Cartesian: observed order (finest pair)", "value": 1.910899908938209, "expected": ">= 1.75 (design rate p + 1 = 2, tolerance 0.25)", "passed": true}, {"name": "p = 2, Cartesian: observed order (finest pair)", "value": 2.969188638966348, "expected": ">= 2.75 (design rate p + 1 = 3, tolerance 0.25)", "passed": true}, {"name": "p = 3, Cartesian: observed order (finest pair)", "value": 3.979401717976675, "expected": ">= 3.75 (design rate p + 1 = 4, tolerance 0.25)", "passed": true}, {"name": "p = 1, curved: observed order (finest pair)", "value": 1.8730120798847445, "expected": ">= 1.35 (empirical floor p + 0.35 for coarse warped meshes; guaranteed rate 1, design rate 2)", "passed": true}, {"name": "p = 2, curved: observed order (finest pair)", "value": 2.597129699007224, "expected": ">= 2.35 (empirical floor p + 0.35 for coarse warped meshes; guaranteed rate 2, design rate 3)", "passed": true}, {"name": "p = 3, curved: observed order (finest pair)", "value": 3.8249372383226583, "expected": ">= 3.35 (empirical floor p + 0.35 for coarse warped meshes; guaranteed rate 3, design rate 4)", "passed": true}], "charts": [{"id": "error_vs_h", "type": "convergence", "title": "Error of the conserved state", "xlabel": "h", "ylabel": "L2 error", "xscale": "log", "yscale": "log", "series": [{"name": "p = 1, Cartesian", "x": [0.25, 0.125, 0.0625], "y": [0.07532555775196094, 0.020975104776561823, 0.005577838654386249]}, {"name": "p = 2, Cartesian", "x": [0.25, 0.125, 0.0625], "y": [0.004486355123781006, 0.0005893957721339443, 7.526484526629533e-05]}, {"name": "p = 3, Cartesian", "x": [0.25, 0.125, 0.0625], "y": [0.00018968477853175222, 1.2419487029034063e-05, 7.87379994769587e-07]}, {"name": "p = 1, curved", "x": [0.25, 0.125, 0.0625], "y": [0.08098661157570344, 0.022898609813681473, 0.006251385752453233]}, {"name": "p = 2, curved", "x": [0.25, 0.125, 0.0625], "y": [0.007430040401262884, 0.001268284868238158, 0.0002096055939625693]}, {"name": "p = 3, curved", "x": [0.25, 0.125, 0.0625], "y": [0.0011703628418926041, 9.695258123242038e-05, 6.841298433919801e-06]}], "reference_slopes": [{"slope": 2, "label": "h^2"}, {"slope": 3, "label": "h^3"}, {"slope": 4, "label": "h^4"}], "note": "Observed order minus the design rate p + 1 on the finest mesh pair: Cartesian -0.09 to -0.02, curved -0.40 to -0.13. On the curved meshes the observed order increased with every refinement for every degree."}], "tables": [{"id": "orders", "title": "Errors and observed orders", "columns": ["mesh", "p", "N", "L2 error", "order"], "rows": [["Cartesian", 1, 4, 0.07532555775196094, null], ["Cartesian", 1, 8, 0.020975104776561823, 1.844461433017249], ["Cartesian", 1, 16, 0.005577838654386249, 1.910899908938209], ["Cartesian", 2, 4, 0.004486355123781006, null], ["Cartesian", 2, 8, 0.0005893957721339443, 2.928235206667834], ["Cartesian", 2, 16, 7.526484526629533e-05, 2.969188638966348], ["Cartesian", 3, 4, 0.00018968477853175222, null], ["Cartesian", 3, 8, 1.2419487029034063e-05, 3.9329264214738786], ["Cartesian", 3, 16, 7.87379994769587e-07, 3.979401717976675], ["curved", 1, 4, 0.08098661157570344, null], ["curved", 1, 8, 0.022898609813681473, 1.8224234121677423], ["curved", 1, 16, 0.006251385752453233, 1.8730120798847445], ["curved", 2, 4, 0.007430040401262884, null], ["curved", 2, 8, 0.001268284868238158, 2.55049123130807], ["curved", 2, 16, 0.0002096055939625693, 2.597129699007224], ["curved", 3, 4, 0.0011703628418926041, null], ["curved", 3, 8, 9.695258123242038e-05, 3.593532751984995], ["curved", 3, 16, 6.841298433919801e-06, 3.8249372383226583]]}], "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": "2026-11-12T10:27:38-0500"}, "wall_time": 212.49, "level": "quick"}