Multidimensional Summation-By-Parts Operators: General Theory and\n Application to Simplex Elements
Bibliographic record
Abstract
Summation-by-parts (SBP) finite-difference discretizations share many\nattractive properties with Galerkin finite-element methods (FEMs), including\ntime stability and superconvergent functionals; however, unlike FEMs, SBP\noperators are not completely determined by a basis, so the potential exists to\ntailor SBP operators to meet different objectives. To date, application of\nhigh-order SBP discretizations to multiple dimensions has been limited to\ntensor product domains. This paper presents a definition for multi-dimensional\nSBP finite-difference operators that is a natural extension of one-dimensional\nSBP operators. Theoretical implications of the definition are investigated for\nthe special case of a diagonal norm (mass) matrix. In particular, a\ndiagonal-norm SBP operator exists on a given domain if and only if there is a\ncubature rule with positive weights on that domain and the polynomial-basis\nmatrix has full rank when evaluated at the cubature nodes. Appropriate\nsimultaneous-approximation terms are developed to impose boundary conditions\nweakly, and the resulting discretizations are shown to be time stable. Concrete\nexamples of multi-dimensional SBP operators are constructed for the triangle\nand tetrahedron; similarities and differences with spectral-element and\nspectral-difference methods are discussed. An assembly process is described\nthat builds diagonal-norm SBP operators on a global domain from element-level\noperators. Numerical results of linear advection on a doubly periodic domain\ndemonstrate the accuracy and time stability of the simplex operators.\n
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How this classification was reachedexpand
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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".