Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system
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Bibliographic record
Abstract
This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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