Grid convergence of the three-dimensional guiding-center construction

Determine whether the three-dimensional guiding-center residual-minimization construction achieves grid convergence under refinement, including product-rule refinement, cross-grid positivity, and refinement-stable moments.

Background

The paper evaluates precise-QA, HSX, and W7-X distributions on a common finite three-dimensional grid using an augmented conservative guiding-center residual and a fixed Polak–Ribie8re-plus iteration budget. Although the base-grid calculations meet the prescribed residual target, the corresponding common coarse/base/fine refinement study does not complete: two fine-grid batches remain above the target after the maximum of 10,000 iterations, so a fine-grid endpoint is not defined under the stated stopping protocol.

Consequently, the paper does not establish convergence of the residual-minimization results with grid refinement. The unresolved scope explicitly includes product-rule refinement, positivity across grids, and stability of computed moments under refinement.

References

Grid convergence, product-rule refinement, cross-grid positivity, refinement-stable moments, and the constrained three-dimensional endpoint remain unexecuted. We therefore report grid convergence as unresolved rather than infer it from the successful base-grid endpoint.

— Gradient-Based Construction of Collisionless Steady-State Guiding-Center Distributions in Tokamaks and Stellarators  (2609.19953 - Yu et al., 17 Sep 2026) in Section “Grid-refinement limitations”