Modified Entropy Correction Formula for the Roe Scheme
Bibliographic record
Abstract
1To avoid un-realistic solutions like expansion shocks from appearing as a part of a solution it is necessary to satisfy the entropy condition for the Roe scheme. A vari-ety of entropy fix formulae for the Roe scheme have been addressed in the literature. Three of the most famous are due to Harten-Hyman and Hoffmann-Chiang. These for-mulations have been assessed in this paper by applying them to the inviscid Burgers ’ equation and shock tube problem. These entropy fix formulations are unable to totally diminish the expansion shock in the vicinity of sonic expansions. Moreover, they are not universal, i.e. a single formulation is not adequate for the scalar Burg-ers ’ equation, shock tube problem and multi-dimensional cases and different formulations were suggested for each case. A simple modification to the Harten formulation is presented in this paper. This modification basically enlarges the band over which the entropy fix condition is enforced. The resulting formulation is able to totally remove the non-physical expansion shocks from the re-gion of sonic expansion without affecting the rest of the computational domain. Comparison among the exact solution, and the entropy correction formulae of Harten-Hyman and Hoffmann-Chiang and the currently mod-ified formula are shown here. The modified entropy fix formulation can totally diffuse the expansion shock. Moreover, the current formula does not affect the solu-tion in the rest of the computational domain. Besides the modified formula, as a single formulation, can uni-versally be applied over a wide range of applications from scalar equations to the governing equation of fluid mo-tion. Finally in this paper, the following test cases are performed to assess the accuracy of the modified entropy formulation: inviscid shear flow, transonic flow over a bump, and transonic flow in a Laval Nozzle. Very accu-rate results are obtained. In the current study a second order upwind scheme of Roe with minmod flux limiter is applied. ∗Ph.D. Candidate, AIAA student member.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".