Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice momentum equation
Bibliographic record
Abstract
We introduce the preconditioned generalized minimum residual (GMRES) method, along with an outer loop (OL) iteration to solve the sea‐ice momentum equation. The preconditioned GMRES method is the linear solver. GMRES together with the OL is used to solve the nonlinear momentum equation. The GMRES method has low storage requirements, and it is computationally efficient and parallelizable. It was found that the preconditioned GMRES method is about 16 times faster than a stand‐alone successive overrelaxation (SOR) solver and three times faster than a stand‐alone line SOR (LSOR). Unlike stand‐alone SOR and stand‐alone LSOR, the cpu time needed by the preconditioned GMRES method for convergence weakly depends on the relaxation parameter when it is smaller than the optimal value. Results also show that with a 6‐hour time step, the free drift velocity field is a better initial guess than the previous time step solution. For GMRES, the symmetry of the system matrix is not a prerequisite. The Coriolis term and the off‐diagonal part of the water drag term can then be treated implicitly. The implicit treatment eliminates an instability characterized by a residual oscillation in the total kinetic energy of the ice pack that can be present when these off‐diagonal terms are handled explicitly. Treating these terms explicitly prevents one from obtaining a high‐accuracy solution of the sea‐ice momentum equation unless a corrector step is applied. In fact, even after a large number of OL iterations, errors in the drift of the same magnitude as the drift itself can be present when these terms are treated explicitly.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| 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".