High-Order Compact-Stencil Summation-By-Parts Operators for the Compressible Navier-Stokes Equations
Bibliographic record
Abstract
A general framework is presented for deriving minimum-stencil high-order summationby-parts finite-difference operators for the second derivative with variable coefficients, for orders of accuracy 3 through 6. These operators can be used to construct time-stable numerical schemes with simultaneous approximation terms to weakly impose boundary conditions. The derivation of these operators leads to various free parameters which can be used for optimization of the operator about criteria such as spectral radius and truncation error. The operators are 2p accurate on the interior, where the interior stencil has 2p+1 nodes, but p accurate at the boundaries. Nonetheless, for purely parabolic problems, they can be shown to be p+2 globally accurate. However, for the Navier-Stokes equations, the continuity equation renders the method p + 1 accurate. We present a novel means of circumventing this degradation in accuracy by using p + 2 globally accurate operators for the continuity equation, and we prove that the new discretization remains amenable to the energy method, a necessary condition to prove time stability. Numerical tests on the one-dimensional linear convection-diffusion equation and the oneand three-dimensional Navier-Stokes equations using the method of manufactured solutions are used for verification and characterization studies. We show that for the Navier-Stokes equations using a p+ 2 globally accurate first derivative for the continuity equation substantially increases the accuracy benefits of the minimum-stencil operator.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".