We present a novel numerical method for solving the anisotropic diffusion
equation in toroidally confined magnetic fields which is efficient, accurate
and provably stable. The continuous problem is written in terms of a derivative
operator for the perpendicular transport and a linear operator, obtained
through field line tracing, for the parallel transport. 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 and asymptotic preserving. Discrete energy estimates are
shown to match the continuous energy estimate given the correct choice of
penalty parameters. Convergence tests are shown for the perpendicular operator
by itself, and the ``NIMROD benchmark" problem is used as a manufactured
solution to show the full scheme converges even in the case where the
perpendicular diffusion is zero. 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.Comment: 33 pages, 8 figure