Fully-Discrete Analysis of High-Order Spatial Discretizations with Optimal Explicit Runge–Kutta Methods
Bibliographic record
Abstract
High-order unstructured methods have become a popular choice for the simulation of complex unsteady flows. Flux reconstruction (FR) is a high-order spatial discretization method, which has been found to be particularly accurate for scale-resolving simulations of complex phenomena. In addition, it has been shown to provide sufficient dissipation for implicit large-eddy simulation (ILES). In conjunction with an FR discretization, an appropriate temporal scheme must be chosen. A common choice is explicit schemes due to their efficiency and ease of implementation. However, these methods usually require a small time-step size to remain stable. Recently, the development of optimal explicit Runge–Kutta (OERK) schemes has enabled stable simulations with larger time-step sizes. Hence, we analyze the fully-discrete properties of the FR method with OERK temporal schemes. We show results for first, second, third, fourth and eighth-order OERK schemes. We observe that OERK schemes modify the spectral behaviour of the semidiscretization. In particular, dissipation decreases in the region of high wavenumbers. We observe that higher-order OERK schemes require a smaller time step than the low-order schemes. However, they follow the dispersion relations of the FR scheme for a larger range of wavenumbers. We validate our analysis with simple advection test cases. It was observed that first and second-degree temporal schemes introduce a relatively large amount of error in the solutions. A one-dimensional ILES test case showed that, as long as the time-step size is not in the vicinity of the stability limit, results are generally similar to classical RK schemes.
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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 | 0.005 |
| 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 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".