Relaxing the CFL Number of the Discontinuous Galerkin Method
Bibliographic record
Abstract
We propose a family of high order methods for the solution of hyperbolic conservation laws which are based on the discontinuous Galerkin (DG) spatial discretization. In the standard DG method, the dispersion and dissipation errors and the spectrum of the semidiscrete scheme are related to the $[\frac{p}{p+1}]$ Padé approximants of $\exp(z)$ and $\exp(-z)$. These Padé approximants are responsible for the superconvergent $\mathcal{O}(h^{2p+2})$ and $\mathcal{O}(h^{2p+1})$ errors in dispersion and dissipation, respectively, and the restriction of the CFL number when increasing the order of approximation, $p$. By modifying the DG method we obtain different rational approximations of the exponential, thereby sacrificing some of the superconvergence of the method, and construct new schemes which allow larger time steps than the original DG method, while having the same order of convergence in the $\mathcal{L}^2$ norm. This is achieved through modifications to the numerical flux. The schemes preserve the attractive properties of the usual DG method, such as the high order accuracy and compact stencil.
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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 | 0.003 | 0.008 |
| 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 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| 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".