Application of Bound-Preserving Limiters to the Nonlinearly Stable Flux Reconstruction High-Order Method
Bibliographic record
Abstract
The flux reconstruction method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the Discontinuous Galerkin (DG) method, on unstructured grids over complex geometries. Under a class of energy stable flux reconstruction (ESFR) schemes, the flux reconstruction method allows for larger time-steps than DG while ensuring stability for linear advection on linear elements. For nonlinear problems, split forms and entropy-conserving flux differencing approaches have become popular since they guarantee robustness for unsteady problems on coarse unstructured grids. Nonlinearly stable flux reconstruction (NSFR) combines the key properties of provable nonlinear stability and the increased time-step from ESFR. NSFR has successfully been applied to unsteady compressible flows in arbitrary curvilinear coordinates while utilizing low-storage weight-adjusted approaches to scale efficiently with low memory consumption. Unfortunately, these schemes fail to satisfy the maximum principle for scalar conservation laws nor do they preserve positivity in the case of hyperbolic conservation laws. This paper incorporates bound-preserving limiters within an NSFR framework to obtain a robust solution that preserves the desired properties while maintaining a high-order of accuracy. This is verified with results for the 1D Burgers' Equation, 2D Burgers' Equation, 2D Linear Advection, 1D Sod Shock Tube, 1D Leblanc Shock Tube and 2D Low Density 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 | 0.001 | 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 | 0.002 |
| 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".