A high-order non field-aligned approach for the discretization of strongly anisotropic diffusion operators in magnetic fusion

In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented.