Discrete Approximation of a Nonlinear Diffusion Equation With Neumann Boundary Conditions: Numerical Study of Quenching Phenomenon
Bibliographic record
Abstract
This paper is concerned with the study of the numerical approximation for  the following nonlinear diffusion equation $ \dfrac{\partial u}{\partial t}=u_{xx} + (1-u)^{-p}, \quad 0<x<1, \quad 0< t< T, $ with a nonlinear singular boundary flux $ u_{x}(0,t)= 0, \quad u_{x}(1,t)=(1-u(1,t))^{-q}, \quad 0< t< T $ and an initial solution $ u(x,0)=u_{0}(x), \quad 0\leq x \leq 1. $ We use the finite differences method to obtain the discrete scheme. Under some conditions, we show that the discrete solution obtained quenches in a finite time and we estimate this quenching time. We also prove that the discrete quenching time converges to the theoretical one when the mesh size tends to zero. Finally, we provide numerical results in tables and present curves and graphs to demonstrate the relevance of our analyses.
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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 | 0.006 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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".