A high-order non field-aligned approach for the discretization of strongly anistropic 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.

本研究提出一种混合间断伽辽金（discontinuous Galerkin）格式，用于求解聚变应用中磁化等离子体环境下产生的极强各向异性扩散问题。相对于主导扩散方向未对齐的非结构网格（unstructured meshes），在离散任意形状几何构型时具备无与伦比的灵活性，但可能引发虚假数值扩散问题。研究采用曲边三角形或四边形单元离散装置的极向平面（poloidal plane），而在环向（toroidal direction）则采用结构化离散格式。数值通量（numerical fluxes）的合理设计，可确保在任意各向异性水平下均能获得准确的收敛阶（convergence order）。经过精心设计的二维与三维数值测试的计算结果表明，只要采用高阶插值，未对齐离散格式即可得到无虚假扩散的求解结果。通过引入数值扩散的显式度量指标，本研究开展了细致的分析，结果显示：数值扩散程度随网格与扩散方向的未对齐程度呈指数增长，同时随插值多项式阶数的提升呈指数下降。本研究还针对两种采用非对齐离散格式且经典应用于聚变建模的有限差分格式（finite-difference schemes），对本方法开展了简要的性能评估。