High-Order Spatial Discretization Methods for the Shallow Water Equations
Bibliographic record
Abstract
We present new numerical methods for the shallow water equations on a sphere in spherical coordinates. In our implementation, the equations are discretized in time with the two-level semi-Lagrangian semi-implicit (SLSI) method, and in space on a staggered grid with the quadratic spline Galerkin (QSG) and the optimal quadratic spline collocation (OQSC) methods. When discretized on a uniform spatial grid, the solutions are shown through numerical experiments to be fourth-order in space locally at the nodes and midpoints of the spatial grids, and third-order globally. We also show that, when applied to a simplified version of the shallow water equations, each of our algorithms yields a neutrally stable solution for the meteorologically significant Rossby waves. Moreover, we demonstrate that the Helmholtz equation associated with the shallow water equations should be derived algebraically rather than analytically in order for the algorithms to be stable with respect to the Rossby waves. These results are verified numerically using Boyd's equatorial wave equations with initial conditions to generate a soliton. We then analyze the performance of our methods on various staggered grids --- the A-, B-, and C-grids. From an eigenvalue analysis of our simplified version of the shallow water equations, we conclude that, when discretized on the C-grid, our algorithms faithfully capture the group velocity of inertia-gravity waves. Our analysis suggests that neither the A- nor B-grids will produce such good results. Our theoretical results are supported by numerical experiments, in which we discretize Boyd's...
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".