Improving the Stability and Efficiency of Operator-Splitting Methods
Bibliographic record
Abstract
Abstract. Operator-splitting methods are widely used to solve differential equations, especially those that arise from multiscale or multiphysics models, because a monolithic (single-method) approach may be inefficient or even infeasible. The most common operator-splitting methods are the first-order Lie–Trotter (or Godunov) and the second-order Strang (Strang–Marchuk) splitting methods. High-order splitting methods with real coefficients require backward-in-time integration in each operator and hence may be adversely impacted by instability for certain operators such as diffusion. However, besides the method coefficients, there are many other ancillary aspects to an overall operator-splitting method that are important but often overlooked. For example, the operator ordering and the choice of subintegration methods can significantly affect the stability and efficiency of an operator-splitting method. In this paper, we investigate some design principles for the construction of operator-splitting methods, including minimization of local error measure, choice of subintegration method, maximization of linear stability, and minimization of overall computational cost. We propose a new four-stage, third-order, 2-split operator-splitting method with seven subintegrations per step and optimized linear stability for a benchmark problem from cardiac electrophysiology. We then propose a general principle to further improve stability and efficiency of such operator-splitting methods by using low-order, explicit subintegrators for unstable subintegrations. We demonstrate an almost 30% improvement in the performance of methods derived from these design principles compared to some well-known high-order methods.
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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 | 0.003 | 0.018 |
| 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 | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".