Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage in Transient Conditions
Bibliographic record
Abstract
ABSTRACT Numerical modeling plays a pivotal role in understanding transient unsaturated flow, which is critical for applications such as groundwater recharge, stormwater management, and contaminant transport. This study investigates the effect of time step refinement on numerical solutions for a vertical infiltration 1D test in a vertical column. First, the element size, ES , was selected to have all calculations in the MCD, the mathematical convergence domain. Then, the numerical and mathematical convergences of the numerical solutions were studied versus the time step, Δ t . The study provided results for hydraulic head, unsaturated hydraulic conductivity, volumetric water content, and vertical water velocity versus elevation, elapsed time t , and Δ t . Asymptotic behavior was obtained for all unknowns when Δ t → 0. All results for all parameters gave linear relationships between the log of the numerical error and the log of Δ t , as predicted by mathematics. Thus, they proved that a good code converges mathematically when ES and Δ t are decreased. For this 1D problem, the MCD is reached for Δ t ≤ 1 s, which is a small MCD. For larger Δ t values (2 ≤ Δ t ≤ 100 s), the code converges numerically in the numerical convergence domain (NCD), where the solutions respect the user‐defined convergence criteria but deviate from the true mathematical convergence criterion, underscoring that a fine temporal discretization is critical. The study also demonstrates that small Δ t steps ensure physically consistent behavior, as reflected in smooth slope volumetric water content versus suction curves, whereas large Δ t steps produce oscillations and instability.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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 source (direct Gemma or distilled Codex), 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".