A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

TL;DR

The paper tackles stable, accurate simulation of time-dependent anisotropic diffusion in strongly magnetized plasmas by formulating a field-aligned diffusion problem in a periodic box and developing a provably stable, SBP-SAT based discretization for the perpendicular term together with a field-line penalty for the parallel term. It introduces an operator-splitting time integration that yields unconditional stability for the fully discrete scheme, supported by discrete energy estimates that mirror the continuous bound. The key contributions include a nonlinear penalty for parallel diffusion, a stable numerical parallel map via field-line tracing, and demonstration of asymptotic preserving behavior against large parallel diffusion on benchmarks such as NIMROD, along with island and chaotic-field tests. The approach enables robust, high-resolution simulations of heat transport in complex magnetic geometries and is implemented in FaADE.jl for reproducibility and further extension to more general coordinates and equilibria.

Abstract

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the ``NIMROD benchmark'' problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

Figures (11)

• Figure 1: Example Poincaré section of a magnetic field in a slab. The feature of the field such as the islands can be controlled by applying a perturbation to the field line Hamiltonian that yields the system of ODEs \ref{['eq:Field line ODE']}. The trajectories of three different field lines are shown.
• Figure 2: Convergence rates of 2D code using method of manufactured solutions for the second order (blue) and fourth order (red) SBP operators. In each case $\Delta t= 0.1 \Delta x$, with $\Delta x=\Delta y$. Top: Spatial convergence. Bottom: Temporal convergence. Labels in the top right of each figure indicated the value of $\theta$ used.
• Figure 3: Exact solution of the "NIMROD benchmark" with $k_\perp=1$ and overlaid with contours and field lines. Field lines (arrows) match the contours of the solution (shown by the solid lines), which implies that regardless of where a field line is terminated along its trajectory it will contribute equally to the parallel diffusion.
• Figure 4: Relative error from decreasing $\Delta x$ due to spatial discretisation. Time-step is scaled as $\Delta t=0.1\Delta x^2$. Top: Odd number of grid points convergence. Bottom: Even number of grid points convergence. Left: Second order spatial discretisation with convergence rate of $\sim2$. Right: Fourth order spatial discretisation with convergence rate of $\sim3.5$.
• Figure 5: Numerical error from decreasing $\Delta t$ as $0.1/2^i$ where $i\in\{0,1,\dots,6\}$ and the grid resolution is fixed at $201\times201$. Left: Second order spatial discretisation.. Right: Fourth order spatial discretisation.

Theorems & Definitions (21)

• Theorem 2.1
• Remark
• Remark
• Theorem 3.1
• Remark
• Remark
• Remark
• Remark
• Remark
• Lemma 3.2