An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation

We propose a semi-Lagrangian numerical algorithm for a time-dependent, anisotropic temperature transport equation in magnetized plasmas in regimes with negligible variation of the magnitude of the magnetic field B along field lines. The approach is based on a formal integral solution of the parallel (i.e., along the magnetic field) transport equation with sources. While this study focuses on a Braginskii (local) heat flux closure, the approach is able to accommodate nonlocal parallel heat flux closures as well. The numerical implementation is based on an operator-split formulation, with two straightforward steps: a perpendicular transport step (including sources), and a Lagrangian (field-line integral) parallel transport step. Algorithmically, the first step is amenable to the use of modern iterative methods, while the second step has a fixed cost per degree of freedom (and is therefore algorithmically scalable). Accuracy-wise, the approach is free from the numerical pollution introduced by the discrete parallel transport term when the perpendicular to parallel transport coefficient ratio _{χ ⊥}/_χ∥ becomes arbitrarily small, and is shown to capture the correct limiting solution when ε=χ⊥L∥2χ∥L⊥2→0 (with _L∥, _{L ⊥} the parallel and perpendicular diffusion length scales, respectively). Therefore, the approach is asymptotic-preserving. We demonstrate the performance of the scheme with several numerical experiments with varying magnetic field complexity in two dimensions, including the case of heat transport across a magnetic island in cylindrical geometry in the presence of a large guide field.