Development and Investigation of Accurate High-order Generalized Summation-by-parts Discretizations for Computational Fluid Dynamics
Bibliographic record
Abstract
The numerical solution of the equations governing turbulent fluid flows, whether the Reynolds-averaged Navier-Stokes equations or other approaches involving the Navier-Stokes equations, as in direct and large-eddy simulations, is computationally expensive. High-order methods have the potential to reduce this computational cost by providing higher accuracy per degree of freedom relative to low-order schemes. Spatial discretization schemes based on the generalized summation-by-parts property have been developed in recent years as a general approach for designing high-order numerical methods. This thesis presents work that delineates how to obtain accurate solutions and, particularly, superconvergent functionals when solving linear and nonlinear partial differential equations governing computational fluid dynamics problems of increasing practical relevance. The specific focus is on numerical schemes constructed on block-structured grids, where the spatial derivatives in the governing equations are approximated with high-order tensor-product generalized summation-by-parts operators. To begin, various components of high-order flow solvers based on generalized summation-by-parts operators are developed including novel artificial dissipation operators and two approaches for high-order grid generation and refinement for traditional and element-type operators. Next, it is shown that functional superconvergence is retained for generalized summation-by-parts discretizations of the linear convection equation in curvilinear coordinates. Furthermore, four dual-consistent discretizations of the two-dimensional linear convection equation based on the mortar-element and global summation-by-parts-operator approaches are derived and characterized in terms of truncation error, solution accuracy, and functional accuracy. Finally, using information gained from the analysis of the linear convection equation, a generalized summation-by-parts discretization for obtaining superconvergent functionals when solving sufficiently smooth problems governed by the Euler equations is proposed. Furthermore, the key features of a given discretization having an impact on solution and functional accuracy are delineated and analyzed. These features are identified to include the representation of the geometry, the approximation of the metrics, and the approximation of the wall normal in the flow tangency boundary condition.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 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".