COMPARING THE DISCONTINUOUS GALERKIN AND HIGH-ORDER DIAMOND DIFFERENCING METHODS FOR THE TRANSPORT EQUATION ON A LOZENGE-BASED HEXAGONAL GEOMETRY
Bibliographic record
Abstract
This paper presents an implementation and a comparison of two spatial discretisation schemes over a hexagonal geometry for the two-dimensional discrete ordinates transport equation. The methods are a high-order Discontinuous Galerkin (DG) finite element scheme and a high-order Diamond Differencing (DD) scheme. The DG method has been, and is being, studied on the hexagonal geometry, also called a honeycomb mesh – but not the DD method. In this research effort, it was chosen to divide the hexagons into (at least) three lozenges. An affine transformation is then applied onto said lozenges to cast them into the reference quadrilaterals usually studied in finite elements. In practice, this effectively means that the equations used in Cartesian geometry have their terms and operators altered using the Jacobian matrix of the transformation. This was implemented in the discrete ordinates solver of the code DRAGON5. Two 2D benchmark problems were then used for the verification and validation, including one based on the Monju 3D reactor benchmark. It was found that the diamond-differencing scheme seemed better. It converged much faster towards the solution at comparable mesh refinements for first-order expansion of the flux. Even if this difference was not present for second-order, DG was slower, about two to four times slower.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".