Opportunities for efficient high-order methods based on the summation-by-parts property (Invited)
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it