  We show that standard algorithms for anisotropic diffusion based on centered
differencing (including the recent symmetric algorithm) do not preserve
monotonicity. In the context of anisotropic thermal conduction, this can lead
to the violation of the entropy constraints of the second law of
thermodynamics, causing heat to flow from regions of lower temperature to
higher temperature. In regions of large temperature variations, this can cause
the temperature to become negative. Test cases to illustrate this for centered
asymmetric and symmetric differencing are presented. Algorithms based on slope
limiters, analogous to those used in second order schemes for hyperbolic
equations, are proposed to fix these problems. While centered algorithms may be
good for many cases, the main advantage of limited methods is that they are
guaranteed to avoid negative temperature (which can cause numerical
instabilities) in the presence of large temperature gradients. In particular,
limited methods will be useful to simulate hot, dilute astrophysical plasmas
where conduction is anisotropic and the temperature gradients are enormous,
e.g., collisionless shocks and disk-corona interface.
