Monolithic multigrid methods for high-order \ndiscretizations of time-dependent PDEs
Bibliographic record
Abstract
A currently growing interest is seen in developing solvers that couple high-fidelity and \nhigher-order spatial discretization schemes with higher-order time stepping methods \nfor various time-dependent fluid plasma models. These problems are famously known \nto be stiff, thus only implicit time-stepping schemes with certain stability properties \ncan be used. Of the most powerful choices are the implicit Runge-Kutta methods \n(IRK). However, they are multi-stage, often producing a very large and nonsymmetric \nsystem of equations that needs to be solved at each time step. There have been recent \nefforts on developing efficient and robust solvers for these systems. We have accomplished \nthis by using a Newton-Krylov-multigrid approach that applies a multigrid \npreconditioner monolithically, preserving the system couplings, and uses Newton’s \nmethod for linearization wherever necessary. We show robustness of our solver on the \nsingle-fluid magnetohydrodynamic (MHD) model, along with the (Navier-)Stokes and \nMaxwell’s equations. For all these, we couple IRK with higher-order (mixed) finiteelement \n(FEM) spatial discretizations. In the Navier-Stokes problem, we further \nexplore achieving more higher-order approximations by using nonconforming mixed \nFEM spaces with added penalty terms for stability. While in the Maxwell problem, \nwe focus on the rarely used E-B form, where both electric and magnetic fields are \ndifferentiated in time, and overcome the difficulty of using FEM on curved domains \nby using an elasticity solve on each level in the non-nested hierarchy of meshes in the \nmultigrid method.
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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) | 0.001 | 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 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".