Tensor-Product Summation-by-Parts Operators
Bibliographic record
Abstract
This paper presents a numerical investigation of the tradeoffs between various discretization approaches and operators, based on diagonal-norm summation-by-parts (SBP) operators, using the two-dimensional linear convection equation and simultaneous approximation terms (SATs) for the weak imposition of boundary conditions and interface coupling. In particular, it focuses on operators which include boundary nodes. Of the operators considered, the hybrid-Gauss-trapezoidal-Lobatto SBP operators are the most efficient. Little difference in efficiency is observed between the divergence and skew-symmetric forms, making the latter preferred given its provable stability on curved meshes. The traditional finite-difference refinement strategy is the most efficient, and the discontinuous element approach the least. The continuous element refinement strategy has comparable efficiency to the traditional approach when not exhibiting lower convergence rates. This motivates a hybrid approach whereby discontinuous elements are constructed from continuous subelements. This hybrid approach is found to inherit the higher convergence rates of the traditional and discontinuous approaches, and higher efficiency relative to the discontinuous approach.
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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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
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".