On variable step highly stable 4-stage Hermite-Birkhoff solvers for stiff ODEs
Bibliographic record
Abstract
Variable-step (VS) \(4\)-stage \(k\)-step Hermite--Birkhoff (HB) methods of order \(p=(k+1)\), denoted by HB\((p)\), are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a two-step diagonally implicit \(4\)-stage Runge--Kutta method of order 3 (TSDIRK3) for solving stiff ordinary differential equations. The main reason for considering this class of formulae is to obtain a set of \(k\)-step methods which are highly stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The approach, described in the present paper, allows us to develop \(L\)-stable \(k\)-step methods of order up to 7 and \(L(\alpha)\)-stable methods of order up to 10 with \(\alpha > 64^\circ\). Fast algorithms are developed for solving confluent Vandermonde-type systems of the new methods in O\((p^2)\) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The step sizes of these methods are controlled by a local error estimator. Selected HB(\(p\)) of order \(p\), \(p=4,5,\ldots,9\), compare favorably with existing Cash modified extended backward differentiation formulae, MEBDF(\(p\)), \(p=4,5,\ldots,8\) in solving problems often used to test highly stable stiff ODE solvers on the basis of CPU time, number of steps and error at the endpoint of the integration interval.
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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 | 0.003 |
| 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 | 0.001 |
| 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".