Numerical solution of the viscous Burgers’ equation using Localized Differential Quadrature method
Bibliographic record
Abstract
In this article, we propose a numerical method to solve the viscous Burgers’ equation in one and two dimensions using the Localized Differential Quadrature (LDQ) scheme. To approximate spatial derivatives, we use the differential quadrature technique locally in a neighborhood of each node using Lagrangian weights and then march the numerical solution in time using the Taylor’s series approximation. Convergence analysis is done by observing that the error between the numerical and exact solutions goes to zero as the number of subdivisions of the spatial domain increases. The localization technique of the differential quadrature method maintains the stability and accuracy of our proposed numerical scheme. We test our proposed numerical scheme using LDQ for several examples. In comparison with several other known methods in the literature, our numerical results confirm the effectiveness of our proposed method and demonstrate their advantages over other known methods in terms of accuracy and computational complexity.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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 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".