A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation.

Large transport anisotropy (χ_parallel/χ_{perpendicular} ∼10¹⁰), chaotic magnetic fields, and non-local heat closures make solving the electron transport equation in magnetized plasmas extremely challenging. A recently developed asymptotic-preserving semi-Lagrangian method¹ overcomes this complexity by an analytical treatment of the direction parallel to the magnetic field in conjunction with modern preconditioning for perpendicular direction. In principle, the method is able to deal with arbitrary anisotropy ratios, different parallel heat-flux closures, and non-trivial magnetic topologies accurately and efficiently. However, the approach was first-order operator-split, and featured an accuracy-based time step limitation, which can be problematic in the presence of islands, and stochastic regions. Here, we present the extension of this algorithm to allow implicit time integration. The implicit algorithm is second-order accurate, and guarantees superior conservation and positivity-preserving properties, which were not ensured by the operator-split implementation. We demonstrate the merits and accuracy of the method with a two dimensional boundary layer problem, which admits an exact analytical solution.

[1] - L. Chacon, et al., JCP, 272, 719, 2014