An efficient method for the anisotropic diffusion equation in magnetic fields
Authors: Muir Dean , Duru Kenneth , Hole Matthew , Hudson Stuart 
Published: October 2023

We solve the anisotropic diffusion equation in 2D, where the dominant direction of diffusion is defined by a vector field which does not conform to a Cartesian grid. Our method uses operator splitting to separate the diffusion perpendicular and parallel to the vector field. The slow time scale is solved using a provably stable finite difference formulation in the perpendicular to the vector field, and an integral operator for the diffusion parallel to it. Energy estimates are shown to for the continuous and semi-discrete cases. Numerical experiments are performed showing convergence of the method, and examples is given to demonstrate the capabilities of the method.
References

L. Chacón, D. del-Castillo-Negrete, and C. D. Hauck. An asymptotic-preserving semi-Lagrangian algorithm for the time-dependent anisotropic heat transport equation. J. Comput. Phys. 272 (2014), pp. 719–746. doi: 10.1016/j.jcp.2014.04.049
D. C. Del Rey Fernández, J. E. Hicken, and D. W. Zingg. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95 (2014), pp. 171–196. doi: 10.1016/j.compfluid.2014.02.01
K. Duru and K. Virta. Stable and high order accurate difference methods for the elastic wave equation in discontinuous media. J. Comput. Phys. 279 (2014), pp. 37–62. doi: 10.1016/j.jcp.2014.08.04
R. Fitzpatrick. Helical temperature perturbations associated with tearing modes in tokamak plasmas. Phys. Plasmas 2.3 (1995), pp. 825–838. doi: 10.1063/1.87143
S. Günter, Q. Yu, J. Krüger, and K. Lackner. Modelling of heat transport in magnetised plasmas using non-aligned coordinates. J. Comput. Phys. 209.1 (2005), pp. 354–370. doi: 10.1016/j.jcp.2005.03.021
P. Helander, S. R. Hudson, and E. J. Paul. On heat conduction in an irregular magnetic field. Part 1. J. Plasma Phys. 88.1, 905880122 (Feb. 2022). doi: 10.1017/S002237782100129X
S. R. Hudson and J. Breslau. Temperature contours and ghost surfaces for chaotic magnetic fields. Phys. Rev. Lett. 100.9, 095001 (2008). doi: 10.1103/PhysRevLett.100.095001
K. Mattsson. Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51.3 (2012), pp. 650–682. doi: 10.1007/s10915-011-9525-
K. Mattsson and J. Nordström. Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199.2 (2004), pp. 503–540. doi: 10.1016/j.jcp.2004.03.001
J. Nordström and T. Lundquist. Summation-by-parts in time: The second derivative. SIAM J. Sci. Comput. 38.3 (2016), A1561–A1586. doi: 10.1137/15M103861X
E. J. Paul, S. R. Hudson, and P. Helander. Heat conduction in an irregular magnetic field. Part 2. Heat transport as a measure of the effective non-integrable volume. J. Plasma Phys. 88.1, 905880107 (2022). doi: 10.1017/S002237782100130
P. J. Roache. Code verification by the method of manufactured solutions. J. Fluids Eng. 124.1 (2002), pp. 4–10. doi: 10.1115/1.1436090
S. Steinberg and P. J. Roache. Symbolic manipulation and computational fluid dynamics. J. Comput. Phys. 57.2 (1985), pp. 251–284. doi: 10.1016/0021-9991(85)90045-
M. Svärd and J. Nordström. Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268 (2014), pp. 17–38. doi: 10.1016/j.jcp.2014.02.03
D. del-Castillo-Negrete and L. Chacón. Local and nonlocal parallel heat transport in general magnetic fields. Phys. Rev. Lett. 106.19, 195004 (2011). doi: 10.1103/PhysRevLett.106.19500
D. del-Castillo-Negrete and L. Chacón. Parallel heat transport in integrable and chaotic magnetic fields. Phy. Plasmas 19.5, 056112 (2012). doi: 10.1063/1.369605