A Modified Preconditioning Approach for Nodal Integral Method
Bibliographic record
Abstract
Nodal Integral Methods (NIM) are numerical methods which solves partial differential equations in quite efficient and accurate manner. The available conventional methods such as Finite Difference Method (FDM) or Finite Volume Methods (FVM) need very fine mesh compared to NIM to attain the same level of accuracy. The high accuracy of NIM is due to the use of semi analytic solutions of approximate ODEs for each node in the mesh. In spite of significant merits over other methods, the prevailing use of NIM is limited only for linear or weakly nonlinear problems. Recently, some efforts have been made to extend the application of NIM for higher non-linearity by using Jacobian-Free Newton Krylov (JFNK) approach which is an efficient implementation of the Newton method. A Preconditioning algorithm of JFNK is given for Burger's equation in both dimension in 1D and 2D, but the work was not extended for higher nonlinearity because of the singularities in the coefficients at higher Reynolds numbers. In the present study, some modifications in the coefficients are given to extend the algorithm for relatively high nonlinearity compared to earlier work. The numerical results are compared with the analytical solution to demonstrate the accuracy of the proposed algorithm. Furthermore, the spectral analysis is performed to test the eigenvalues clustering ability of the proposed algorithm.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".