AI summary · arXiv: 2306.00423v1 · math.NA
Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fields
The essentials
- The authors present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields.
- The method is efficient, accurate and provably stable.
- 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 operators obtained by field line tracing.
- The fully-discrete approximation is proven to be unconditionally stable and asymptotic preserving.
- Convergence tests are shown for the perpendicular operator by itself, and a "NIMROD benchmark" problem is used as a manufactured solution to demonstrate that the full scheme converges even when perpendicular diffusion is zero.
- An example magnetic field with chaotic regions and islands is presented to show that contours of the anisotropic diffusion equation reproduce key features in the field.
- A Poincaré section provides context on this example magnetic field by highlighting three different field line trajectories; two stay close to "magnetic surfaces" while one lies in a chaotic region resulting in large excursions in ψ values; there is also a trajectory with distinct "holes" which may allow nearby field lines to pass through.
- This paper provides an important contribution to numerical methods for solving anisotropic diffusion equations in toroidally confined magnetic fields, with applications to plasma physics research.
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