Relaxing the CFL Number of the Discontinuous Galerkin Method
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it