Opportunities for efficient high-order methods based on the summation-by-parts property (Invited)
Bibliographic record
Abstract
Summation-by-parts (SBP) operators are traditionally viewed as high-order finite-difference operators, but they can also be interpreted as finite-element operators with an implicit basis. Such an element-based perspective leads to several opportunities that we describe. The first is provided by generalized one-dimensional SBP operators, which maintain the desirable properties of classical SBP operators while permitting flexible nodal distributions. The second opportunity is to extend the SBP definition to multiple dimensions, and a recently proposed definition for multidimensional SBP operators paves the way for time-stable, high-order finite-difference operators on unstructured grids. The final opportunity that we discuss is an analogy with the continuous Galerkin finite-element method, which leads to a systematic means of assembling SBP operators on a global domain from elemental operators. To illustrate these ideas, high-order SBP operators are constructed for the triangle and tetrahedron, and the former are assembled into a global SBP operator and applied to a linear convection problem on a triangular grid.
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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 | 0.002 | 0.003 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".