Stability of Leapfrog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations: Flat-Terrain Case
Bibliographic record
Abstract
The stability of semi-implicit schemes for the hydrostatic primitive equations system has been studied extensively over the past 20 yr, since this temporal scheme and this system represented a standard for NWP.However, with the increase of computational power, the relaxation of the hydrostatic approximation through the use of nonhydrostatic fully elastic systems is now emerging for future NWP as an attractive solution valid at any scale.In this context, several models employing the so-called Euler equations together with a constantcoefficients semi-implicit time discretization have already been developed, but no solid justification for the suitability of this algorithmic combination has been presented so far, especially from the point of view of robustness.The aim of this paper is to investigate the response of this system/scheme in terms of stability in presence of explicitly treated residual terms, as it inevitably occurs in the reality of NWP.This study is restricted to the impact of thermal and baric residual terms (metric residual terms linked to the orography are not considered here).It is shown that, conversely to what occurs with hydrostatic primitive equations, the choice of the prognostic variables used to solve the system in time is of primary importance for the robustness with Euler equations.For an optimal choice of prognostic variables, unconditionally stable schemes can be obtained (with respect to the length of the time step), but only for a smaller range of reference states than in the case of hydrostatic primitive equations.This study also indicates that (i) vertical coordinates based on geometrical height and on mass behave similarly in terms of stability for the problems examined here, and (ii) hybrid coordinates induce an intrinsic instability, the practical importance of which is, however, not completely elucidated in the theoretical context of this paper.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".