Part B: Finite-Difference Methods for Extremely Anisotropic Diffusion
Pp. 29-52 (24)
Bram van Es, Barry Koren and Hugo J. de Blank
Anisotropic diffusion is a common physical phenomenon and describes
processes where the diffusion of some scalar quantity is directionally dependent.
Anisotropic diffusive processes are for instance Darcy’s flow for porous media, large
scale turbulence where turbulence scales are anisotropic in size, and heat conduction
and momentum dissipation in fusion plasmas. In fusion plasmas there is extreme
anisotropy due to the high temperature and large magnetic field strength. This causes
diffusive processes, heat diffusion and energy/momentum loss due to viscous friction, to
effectively be aligned with the magnetic field lines. This alignment leads to different
values for the respective diffusive coefficients in the magnetic field direction and in the
perpendicular direction, to the extent that heat diffusion coefficients can be up to 1012
times larger in the parallel direction than in the perpendicular direction. This anisotropy
puts stringent requirements on the numerical methods used to approximate the MHDequations
since any misalignment of the grid may cause the perpendicular diffusion to
be polluted by the numerical error in approximating the parallel diffusion. Currently the
common approach is to apply magnetic field-aligned coordinates, an approach that
automatically takes care of the directionality of the diffusive coefficients. This approach
runs into problems in the case of crossing field lines, e.g., x-points and points where
there is magnetic reconnection. It is therefore useful to consider numerical schemes that
are more tolerant to the misalignment of the grid with the magnetic field lines, both to
improve existing methods and to help open the possibility of applying regular nonaligned
grids. To investigate this, several discretization schemes are applied to the
unsteady anisotropic heat diffusion equation on a cartesian grid. All methods presented
are generic and carry over to any other anisotropic diffusion problem.
Finite-difference discretizations, mimetic discretization, Cartesian
grids, field-aligned coordinates, anisotropic diffusion, perpendicular diffusion,
convergence loss, biquadratic interpolation, tokamak magnetohydrodynamics,
Darcy flow, semi-staggered grid, Vandermonde coefficients, diffusion tensor,
consistency, convergence, numerical diffusion, heat conduction, curvature terms,
modified Euler scheme, angle of misalignment.
Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.