Truncation and Discretization Error for Diffusion Schemes on Unstructured Meshes
Bibliographic record
Abstract
The accuracy and reliability of CFD simulations depend on the ability to reduce and quantify physical modeling and numerical errors. The fact that numerical errors are at least as large as physical modeling errors was highlighted in the results of the 3rd AIAA Drag Prediction workshop, which also showed the increased severity of this issue for unstructured ow solvers. Furthermore, the varied local shape and connectivity of unstructured meshes make it di cult to quantify numerical error. The 5th AIAA Drag Prediction workshop focussed on reducing grid-related errors even further, where a grid re nement study was performed on a common grid sequence derived from a multiblock structured grid. The study had six di erent levels of grid re nement ranging from 136× 10 cells to 0.64× 10 cells, a much larger range than is typically seen, with structured overset and hexahedral, prismatic, tetrahedral, and hybrid unstructured grid formats. The results of the grid re nement study indicated that there was no clear advantage of any one grid type in terms of a reduced scatter in solution. Moreover, there were no clear breakouts with grid type or turbulence model. The conclusion was that discretization errors and turbulence modeling errors are both still major contributors to error in solution. The impact on solution accuracy by the interactions between mesh quality (cell size, shape, and anisotropy) and discretization schemes is not well understood and demands further investigation. The di erence between the discrete operator and the continuous PDE applied to the solution is referred to as the truncation error, while the di erence between the numerically approximated solution and the exact solution is the discretization error. The truncation error can be expressed in terms of the derivatives of an underlying smooth solution at the points of the discrete domain and can be used to estimate the discretization errors that occur during the approximate numerical solution of PDEs. It can be shown that for the special case of a linear di erential operator L, the truncation error τ can be used to calculate the discretization error e, as
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".