Smooth Polar B-Splines with High-Order Regularity at the Origin
Peiyou Jiang, Roman Hatzky, Zhixin Lu, Eric Sonnendrücker, Matthias Borchardt, Ralf Kleiber, Martin Campos Pinto, Ronald Remmerswaal
Abstract
We introduce a smooth B-spline discretization in polar coordinates on the unit disc that corrects the loss of regularity present at the origin caused by the coordinate singularity in standard tensor-product B-spline formulations. The method constructs “smooth polar splines” via a Galerkin projection of harmonic polar functions S-m (r, θ) ∶= rlsin(mθ) and Sm (r, θ) ∶= rl cos(mθ), derived from the polar representation of Cartesian monomials, onto the central tensor-product B-spline basis in the innermost radial region. The radial component reproduces rl exactly for 0 ≤ l ≤ p, where p is the B-spline degree, satisfying the near-origin regularity condition. However, exact compatibility with C∞ -regularity at the origin is recovered only in the limit Δθ → 0, when the angular component resolves all angular harmonics accurately. The smooth polar splines are linear combinations of standard tensor-product B-splines and lie in the same function space, enabling mapping between the C∞-regular subspace and the original discretization space via an exact prolongation operator and a corresponding restriction operator acting on the discrete variables. They match standard tensor-product B-splines away from the origin, preserve orthogonality among the newly constructed origin-centered basis functions, and maintain local support and sparse matrices. This smoothness and locality improve the conditioning of mass and stiffness matrices, conserve charge, and reduce statistical errors in particle-in-cell simulations near the origin, while eliminating spurious eigenvalues in eigenvalue problems. The approach provides a robust, high-order, and efficient adaptation of tensor-product B-splines for polar coordinates in physics simulations.
BibTeX
@article{Jiang_2026,
title={Smooth Polar B-Splines with High-Order Regularity at the Origin},
url={http://dx.doi.org/10.2139/ssrn.6211977},
DOI={10.2139/ssrn.6211977},
publisher={Elsevier BV},
author={Jiang, Peiyou and Hatzky, Roman and Lu, Zhixin and Sonnendrücker, Eric and Borchardt, Matthias and Kleiber, Ralf and Campos Pinto, Martin and Remmerswaal, Ronald},
year={2026} }