Magneto-Hydrodynamic Eigenvalue Solver for Axisymmetric Equilibria Based on Smooth Polar Splines

We present a new eigenvalue solver for the ideal magneto-hydrodynamics (MHD) equations in axisymmetric equilibria that enables the robust and accurate description of eigenfunctions near the magnetic axis. The algorithm is based on discrete differential forms in combination with C1-smooth polar splines in the framework of isogeometric analysis. The symmetric discretization leads to a Hermitian Alfvén part in the MHD force operator, mirroring the self-adjointness of the continuous operator. Moreover, eigenfunctions are continuous across the magnetic axis by use of the polar spline framework. We provide comparisons to the standard tensor product approach by selected numerical examples and show that a) the polar spline framework correctly reproduces the tensor product results sufficiently far away from the magnetic axis and that b) it gives better results close to the magnetic axis with regards to back transformations to the physical (Cartesian) domain as usually needed if a coupling to a particle-in-cell code is desired.