Computational Algorithms for Solving Spectral/$hp$ Stabilized Incompressible Flow Problems
Bibliographic record
Abstract
In this paper we implement the element-by-element preconditioner and inexact Newton-Krylov methods (developed in the past) for solving stabilized computational fluid dynamics (CFD) problems with spectral methods. Two different approaches are implemented for speeding up the process of solving both steady and unsteady incompressible Navier-Stokes equations. The first approach concerns the application of a scalable preconditioner namely the element by element LU preconditioner, while the second concerns the application of Newton-Krylov (NK) methods for solving non-linear problems. We obtain good agreement with benchmark results on standard CFD problems for various Reynolds numbers. We solve the Kovasznay flow and flow past a cylinder at Re-$100$ with this approach. We also utilize the Newton-Krylov algorithm to solve (in parallel) important model problems such as flow past a circular obstacle in a Newtonian flow field, three dimensional driven cavity, flow past a three dimensional cylinder with different immersion lengths. We explore the scalability and robustness of the formulations for both approaches and obtain very good speedup. Effective implementations of these procedures demonstrate for relatively coarse macro-meshes the power of higher order methods in obtaining highly accurate results in CFD. While the procedures adopted in the paper have been explored in the past the novelty lies with applications with higher order methods which have been known to be computationally intensive.
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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 | 0.000 |
| 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 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".