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

• Published in: Journal of computational physics 2014-09, Vol.272, p.719-746
• Main Authors: Chacón, L., del-Castillo-Negrete, D., Hauck, C.D.
• Format: Article
• Language: English
• Subjects: Algorithms; Anisotropic transport; Anisotropy; Asymptotic preserving methods; Asymptotic properties; Closures; Magnetic fields; Mathematical models; Operator-splitting; Parallel transport; Transport; Transport equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.04.049

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.