Provably Stable Discontinuous Spectral-Element Methods with the Summation-by-Parts Property: Unified Matrix Analysis and Efficient Tensor-Product Formulations on Curved Simplices
Bibliographic record
Abstract
Discontinuous spectral-element methods (DSEMs) provide a flexible high-order spatial discretization approach for time-dependent conservation laws, but are traditionally regarded as lacking robustness relative to conventional second-order numerical methods for such partial differential equations. This thesis describes a unifying matrix analysis framework based on the summation-by-parts (SBP) property for the construction and analysis of DSEMs which are provably stable for linear or nonlinear hyperbolic problems. Within such a framework, we obtain algebraic proofs of conservation and energy stability as well as of the discrete equivalence between strong and weak formulations for discontinuous Galerkin and flux reconstruction schemes on general element types. The proposed methodology is also applied to the development of a new class of efficient and robust DSEMs on triangles and tetrahedra through the construction of sparse tensor-product operators in collapsed coordinates which satisfy the SBP property and are amenable to efficient sum-factorization algorithms. These SBP operators are used to construct split-form and flux-differencing DSEMs on curved triangular and tetrahedral unstructured grids which are conservative, free-stream preserving, and energy-stable for the linear advection equation or entropy stable for nonlinear hyperbolic systems. By exploiting the structure of the proposed operators as well as that of the Proriol-Koornwinder-Dubiner polynomial basis and using a weight-adjusted approximation to avoid the inversion of the curvilinear modal mass matrix, we obtain fully explicit algorithms of reduced complexity which facilitate the efficient extension of provably stable DSEMs on triangles and tetrahedra to arbitrarily high-order accuracy. The proposed schemes are compared to non-tensorial energy- and entropy-stable formulations on triangles and tetrahedra in a series of numerical experiments involving the solution of the linear advection and compressible Euler equations.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".