Discontinuous Galerkin Methods for Partial Differential Equations
Bibliographic record
Abstract
The purpose of this meeting was to bring together researchers in a wide variety of areas working on discontinuous Galerkin (DG) methods for partial differential equations, to investigate and identify problems of current interest and to exchange ideas and viewpoints on the most recent developments of these meth-ods. There were 33 participants, mostly from American and Canadian universities, including students and postdoctoral fellows. The program of the workshop consisted of 28 half-hour talks. 1 Overview of the Field The origins of discontinuous Galerkin methods can be traced back to the seventies where they were introduced as non-standard discretization techniques for the numerical approximation of linear transport equations [1]. A remarkable adavantage of the original DG method is that the approximate solutions can be computed element-by-element when the elements are suitably ordered along the characteristic directions of the transport field. The success of DG methods for linear equations prompted several researchers to try to extend them to non-linear hyperbolic conservation laws. In the early eighties and beginning of the nineties, Cockburn and Shu introduced the Runge-Kutta discontinuous Galerkin (RKDG) methods for scalar conservation laws, see the review article [3] and the references therein. These methods are based on piecewise polynomial space discretizations, combined with total variation diminishing (TVD) explicit time-stepping algorithms. The resulting schemes have several important advantages as compared to, e.g., finite difference methods. The variational structure of DG methods greatly facilitates the handling of complicated geometries and elements of various shapes and types, as well as the treatment of boundary conditions. Moreover, DG mass matrices are block-diagonal and can be inverted at a very low computational cost, giving rise to very efficient time-stepping algorithms. Soon after their introduction, RKDG methods were extended to non-linear hyperbolic systems and to more general convection-diffusions problems. 2
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How this classification was reachedexpand
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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".