A high-temporal-order, asymptotic-preserving spectral deferred correction algorithm for the anisotropic heat transport equation

Modeling electron transport in magnetized plasmas is extremely challenging due to the extreme anisotropy between the parallel (to the magnetic field) and perpendicular directions (the transport-coefficient ratio $\chi_{\parallel}/\chi_{\perp} \sim 10^{10}$ in fusion plasmas). Recently, an asymptotic preserving semi-Lagrangian approach has been developed that is able to deal with arbitrary anisotropy ratios and non-trivial magnetic topologies in an accurate and efficient manner.\footnote{L. Chac\'on, D. del-Castillo-Negrete, C. Hauck, {\em JCP}, submitted (2013)} The approach is shown to avoid spatial discretization pollution, and to feature bounded numerical errors for {\em arbitrary} $\chi_{\parallel}/\chi_{\perp}$ ratios, which renders it asymptotic preserving. However, it is only first-order accurate in time. In this poster, we explore spectral deferred correction (SDC) methods\footnote{A. Dutt, L. Greengard, and V. Rokhlin, {\em BIT} {\bf 40}, 241 (2000)} to produce a high-order asymptotic preserving algorithm, using the first-order semi-Lagrangian algorithm as the inner solver for the corrector step. We will show that the combination SDC+semi-Lagrangian features a numerical stability constraint, but one which is benign for sufficiently large anisotropy ratios.