Quasi-orthogonal Runge-Kutta projection methods
Bibliographic record
Abstract
A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under exact solution of their governing PDEs. However, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce these constraints, leading to a loss of accuracy and stability. Projection is an efficient way to address this shortcoming by correcting the RK solution at the end of each time step. Here we introduce a novel projection method for explicit RK schemes, called a quasi-orthogonal projection method. This method can be employed for systems containing a single (not necessarily convex) invariant functional, for dissipative systems, and for the systems containing multiple invariants. It works by projecting the orthogonal search direction(s) into the solution space spanned by the RK stage derivatives. With this approach linear invariants of the problem are preserved, the time step size remains fixed, additional computational cost is minimal, and these optimal search direction(s) preserve the order of accuracy of the base RK method. This presents significant advantages over existing projection methods. Numerical results demonstrate that these properties are observed in practice for a range of applications.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| 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) | 0.007 | 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".