A High-order Implicit Difference Method for the One-dimensional Convection Diffusion Equation
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Bibliographic record
Abstract
Based on the exponent transform to eliminate the ``convection item'' in the equation and the fourth-order compact difference formulas for the first and second derivatives, two chasses of new implicit difference schemes are proposed for solving the one-dimensional convection-diffusion equation. The methods are of order $O\left({\tau ^2 + h^4} \right)$ and $O\left( {\tau ^4 + h^4} \right)$ respectively. The former is proved to be unconditionally stable while the later is unconditionally unstable by Fourier analysis. The result of numerical experiment shows that the $O\left( {\tau ^2 + h^4} \right)$ scheme is an effective difference scheme to solve the convection diffusion problem.
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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 | 0.009 | 0.006 |
| 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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