Robust and reliable defect control for Runge-Kutta methods
Bibliographic record
Abstract
The quest for reliable integration of initial value problems (IVPs) for ordinary differential equations (ODEs) is a long-standing problem in numerical analysis. At one end of the reliability spectrum are fixed stepsize methods implemented using standard floating point, where the onus lies entirely with the user to ensure the stepsize chosen is adequate for the desired accuracy. At the other end of the reliability spectrum are rigorous interval-based methods, that can provide provably correct bounds on the error of a numerical solution. This rigour comes at a price, however: interval methods are generally two to three orders of magnitude more expensive than fixed stepsize floating-point methods. Along the spectrum between these two extremes lie various methods of different expense that estimate and control some measure of the local errors and adjust the stepsize accordingly. In this article, we continue previous investigations into a class of interpolants for use in Runge-Kutta methods that have a defect function whose qualitative behavior is asymptotically independent of the problem being integrated. In particular the point, in a step, where the maximum defect occurs as h → 0 is known a priori. This property allows the defect to be monitored and controlled in an efficient and robust manner even for modestly large stepsizes. Our interpolants also have a defect with the highest possible order given the constraints imposed by the order of the underlying discrete formula. We demonstrate the approach on three Runge-Kutta methods of orders 5, 6, and 8, and provide Fortran and preliminary Matlab interfaces to these three new integrators. We also consider how sensitive such methods are to roundoff errors. Numerical results for four problems on a range of accuracy requests are presented.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".