A twist-shift boundary condition algorithm for discontinuous Galerkin discretizations

Modern local gyrokinetic simulations account for the turbulent anisotropy ($k_\parallel\sim 1/qR$, $k_\perp\rho\sim 1$) to construct computational grids aligned with the equilibrium magnetic field ($B$). Such technique efficiently distributes degrees of freedom and results in more affordable calculations. These flux-tube simulations have a limited extent along $B$ and impose twist and shift boundary conditions (BCs) [1]. These BCs exploit the small turbulent correlation lengths and the axisymmetry of the device to enforce periodicity in the parallel direction at a fixed poloidal angle. Historically this BC has been employed in pseudo-spectral codes, yet the Gkeyll code [2] uses a discontinuous Galerkin (DG) representation to which the spectral algorithms do not transfer. We thus formulated a DG implementation of twist-shift BCs based on the concept of weak or Galerkin equality and a DG representation of the magnetic safety factor ($q$). Carrying out the necessary integrals in a conservative manner leads us to split up the problem into a series of sub-cell integrals depending on the computational grid and $q$. We present basic tests of the implementation in Gkeyll and a time-dependent local flux-tube simulation.

[1] M. A. Beer, et al., PoP 2, 2687 (1995).

[2] https://gkeyll.readthedocs.io/en/latest/