Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations
Bibliographic record
Abstract
Abstract The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
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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 | 0.006 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".