We extend recently-developed mesh-free Lagrangian methods for numerical magnetohydrodynamics (MHD) to arbitrary anisotropic diffusion equations, including: passive scalar diffusion, Spitzer-Braginskii conduction and viscosity, cosmic ray diffusion/streaming, anisotropic radiation transport, non-ideal MHD (Ohmic resistivity, ambipolar diffusion, the Hall effect), and turbulent 'eddy diffusion.' We study these as implemented in the code GIZMO for both new meshless finite-volume Godunov schemes (MFM/MFV) as well as smoothed-particle hydrodynamics (SPH). We show the MFM/MFV methods are accurate and stable even with noisy fields and irregular particle arrangements, and recover the correct behavior even in arbitrarily anisotropic cases. They are competitive with state-of-the-art AMR/moving-mesh methods, and can correctly treat anisotropic diffusion-driven instabilities (e.g. the MTI and HBI, Hall MRI). We also develop a new scheme for stabilizing anisotropic tensor-valued fluxes with high-order gradient estimators and non-linear flux limiters, which is trivially generalized to AMR/moving-mesh codes. We find that SPH is accurate for isotropic diffusion; however, fundamental errors in the most common SPH discretizations are either numerically unstable, or produce large systematic errors in anisotropic cases that do not converge. We develop a new integral-Godunov SPH formulation which resolves these errors by replacing the standard SPH gradient estimator with a moving least-squares estimator and solving a flux-limited Riemann problem between particles. However in problems with full MHD, well-known SPH errors can still swamp the physical diffusivity.
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