Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations
Bibliographic record
Abstract
It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".