A Parallel Newton-Krylov-Schur Flow Solver for the Navier-Stokes Equations Using the SBP-SAT Approach
Bibliographic record
Abstract
This paper presents a three-dimensional Newton-Krylov flow solver for the Navier-Stokes equations which uses summation-by-parts (SBP) operators on multi-block structured grids. Simultaneous approximation terms (SAT’s) are used to enforce the boundary conditions and the coupling of block interfaces. The discrete equations are solved iteratively with an inexact Newton method. The linear system of each Newton iteration is solved using a Krylov subspace iterative method with an approximate-Schur parallel preconditioner. The algorithm is validated against an established two-dimensional flow solver. Additionally, results are presented for laminar flow around the ONERA M6 wing, as well as low Reynolds number flow around a sphere. Using 384 processors, the solver is capable of obtaining the steady-state solution (reducing the flow residual by 12 orders of magnitude) on a 4.1 million node grid around the ONERA M6 wing in 4.2 minutes. Convergence to 3 significant figures in force coefficients is achieved in 83 seconds. Parallel scaling tests show that the algorithm scales well with the number of processors used. The results show that the SBP-SAT discretization, solved with the parallel Newton-Krylov-Schur algorithm, is an efficient option for three-dimensional Navier-Stokes solutions, with the SAT’s providing several advantages in enforcing boundary conditions and block coupling. I.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".