Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using Compact Numerical Fluxes on Quadrilateral Elements
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. The attraction of the method follows with its stability proofs for the linear advection problem, under a class of energy stable flux reconstruction (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. The proof has later been developed for the diffusion problem on triangular elements for Local Discontinuous Galerkin (LDG) and compact numerical fluxes such as the interior penalty (IP), the Bassi and Rebay II, the compact discontinuous Galerkin, or the compact discontinuous Galerkin 2 numerical fluxes. For the diffusion problem, on Cartesian meshes, the proof has been extended for the LDG numerical flux. This paper expands the proof for compact numerical fluxes, and demonstrates the stability’s independence on the correction parameter in the auxiliary equation for the IP and BR2 numerical fluxes. The conditions for stability restrict the values of the penalty term of the different schemes. These stability conditions are valid for any ESFR schemes including DG and are much sharper than previously known criteria.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 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) | 0.001 | 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".