Numerical Blow-up for a Parabolic Partial Differential Equation With a Convection Term
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Bibliographic record
Abstract
This work numerically analyzes blow-up phenomena in solutions to a parabolic partial differential equation with a convection term, subject to non-homogeneous Neumann boundary conditions and strictly positive initial data. This problem represents a turbulence model of fluid flows. We establish conditions under which the semi-discrete spatial formulation of the continuous problem blows up in finite time and we estimate the corresponding semi-discrete blow-up time. We also prove that the semi-discrete blow-up time converges to real one, when the spatial discretization step tends to zero. From fully discretized schemes, we have developed algorithms that have allowed us to obtain good approximations of the blow-up time and other numerical results to illustrate our analysis.
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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 | 0.007 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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 it