Ronald Remmerswaal

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.

The use of gyrokinetics, wherein phase-space coordinate transformations result in a phase-space dimensionality reduction as well as the removal of fast time scales, has enabled the simulation of microturbulence in fusion devices. The state-of-the-art gyrokinetic models used in practice are parallel-only models wherein the perpendicular part of the vector potential is neglected. Such models are inherently not gauge-invariant. We generalise the work of Burby & Brizard (2019 Phys. Lett. A vol. 383, no. 18, pp. 2172–2175) by deriving a sufficient condition on the gyrocentre coordinate transformation that ensures gauge invariance. This leads to a parametrised family of gyrokinetic models for which we motivate a specific choice of parameters that results in the smallest gyrocentre coordinate transformation for which the resulting gyrokinetic model is consistent, gyro-phase independent, gauge-invariant and has an invariant magnetic moment. Due to gauge invariance, this model can be expressed directly in terms of the electromagnetic fields rather than the potentials, and the gyrokinetic model thereby results in the macroscopic Maxwell’s equations. For the linearised model, it is demonstrated that the shear and compressional Alfvén waves are present with the correct frequencies. The fast compressional Alfvén wave can be removed by making use of a Darwin approximation. This approximation retains the gauge invariance of the proposed model.

For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young-Laplace jump condition. We show that traditional geometric volume of fluid (VoF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems.Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabolic LVIRA and MoF methods, PLVIRA and PMoF, respectively. We show that a Lagrangian remapping method is sufficiently accurate for the advection of such a parabolic interface.It is numerically demonstrated that the newly proposed PPIC methods result in an increase of reconstruction accuracy by one order, convergence of the interface curvature in time-dependent advection problems and Weber number independent convergence of a droplet translation problem, where the advection method is coupled to a two-phase Navier--Stokes solver.