A fully-discrete entropy-stable flux reconstruction scheme for the Eulerequations
Bibliographic record
Abstract
The aerospace industry requires robust, stable, high-fidelity simulations in order to resolve complicated, nonlinear flows. Higher-order methods, such as the discontinuous Galerkin method and flux reconstruction (FR) method, are emerging as the next generation of CFD methods for high-fidelity applications. Entropy stability has gained popularity in higher-order methods over the past decade as a guarantee of nonlinear stability by ensuring the correct evolution of a numerical entropy variable for an arbitrarily long solution time. A method of improving the robustness of higher-order methods is by formulating schemes that ensure entropy stability. They are typically formulated semi-discretely, such that entropy stability is only guaranteed in the spatial variables. Numerical entropy may change due to higher-order temporal integration, so unsteady problems rely on a very small time step size to approximate continuity. However, the added computational cost of entropy-stable split forms and small time step sizes suggests that higher-order entropy-stable methods are not yet feasible for industrial problems. We formulate a fully-discrete entropy-stable schemethat is, one which is entropy stable in spatial and temporal variablesto address the high computational cost of entropystable methods. We use the FR scheme of Cicchino and Nadarajah [1, 2] for the spatial semidiscretization, which is entropy-stable for arbitrary node choices. Temporal entropy stability is addressed through the relaxation Runge-Kutta method The combination allows us to choose a relatively large time step size while retaining the nonlinear stability guarantee. We verify order of accuracy and stability of the fully-discrete, entropy-stable scheme. The properties of the scheme are studied using the Taylor-Green vortex and Kelvin-Helmholtz instability test cases, which are Euler test cases displaying considerable nonlinearity.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| 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".