An asymptotic-preserving semi-Lagrangian algorithm for the anisotropic heat transport equation with arbitrary magnetic fields

TL;DR

This work addresses highly anisotropic heat transport in magnetized plasmas with arbitrary magnetic-field topology. It extends the asymptotic-preserving (AP) semi-Lagrangian scheme from the tokamak-ordering regime to general field compressibility by reformulating along a $\lambda$-coordinate with a positive stabilization field $\beta(\mathbf{x})$ and by using $\mathcal G_{\lambda}$ and $\mathcal P_{\lambda}$ operators that retain the Green's-function structure. The scheme remains stable and consistent, achieving AP behavior in the $\epsilon\to0$ limit and correctly handling magnetic nulls, with convergence demonstrated against a manufactured solution. Numerical tests show at least fourth-order spatial accuracy in non-null configurations and robust performance across a range of $\epsilon$, $B_{0}$, and topology, though singular sources at nulls can cause local order reduction.

Abstract

We extend the recently proposed semi-Lagrangian algorithm for the extremely anisotropic heat transport equation [Chacón et al., J. Comput. Phys., 272 (2014)] to deal with arbitrary magnetic field topologies. The original scheme (which showed remarkable numerical properties) was valid for the so-called tokamak-ordering regime, in which the magnetic field magnitude was not allowed to vary much along field lines. The proposed extension maintains the attractive features of the original scheme (including the analytical Green's function, which is critical for tractability) with minor modifications, while allowing for completely general magnetic fields. The accuracy and generality of the approach are demonstrated by numerical experiment with an analytical manufactured solution.

Figures (5)

• Figure 1: Left panel: flux function with $\delta=0.1$. Right panel: flux function with $\delta=0.5$.
• Figure 2: Mesh convergence study for the tokamak-ordering formulation with a null-space solution ($\tilde{T}=0$) for $\epsilon=10^{-2}$ with $B_{0}=0$ and $\delta=0.1,0.5$. Fourth-order spatial convergence is achieved, as expected.
• Figure 3: Mesh convergence study for the full solution with $\epsilon=10^{-2}$ and $\delta=0.1$. Left: tokamak-ordering formulation for three different guide fields, $B_{0}=0,10,100$, demonstrating convergence of the tokamak-ordering solution to the analytical one as the guide-field $B_{0}$ increases. Right: Comparison of errors between tokamak-ordering and arbitrary-B formulation for $B_{0}=10$, demonstrating the ability of the arbitrary-B formulation to converge where the tokamak-ordering one fails.
• Figure 4: Mesh convergence study with the arbitrary-B formulation for small or zero magnetic guide fields.
• Figure 5: Steady-state errors vs the analytical solution for $\epsilon=10^{-4}$ on the $128\times128$ mesh for various choices of $\delta$ and $B_{0}$.