Phase Space Stability Error Control with Variable Time‐stepping Runge‐Kutta Methods for Dynamical Systems
Bibliographic record
Abstract
Abstract We consider a phase space stability error control for numerical simulation of dynamical systems. We illustrate how variable time‐stepping algorithms perform poorly for long time computations which pass close to a fixed point. A new error control was introduced in [9], which is a generalization of the error control first proposed in [8]. In this error control, the local truncation error at each step is bounded by a fraction of the solution arc length over the corresponding time interval. We show how this error control can be thought of either a phase space or a stability error control. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In particular, we analyze the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. We also consider the dynamics in the neighborhood of saddle points. We introduce a step‐size selection scheme which allows this error control to be incorporated within the standard adaptive algorithm as an extra constraint at negligible extra computational cost. Theoretical and numerical results are presented to illustrate the behavior of this error control. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".