  We present an implementation of thermal conduction including the anisotropic
effects of magnetic fields for SPH. The anisotropic thermal conduction is
mainly proceeding parallel to magnetic fields and suppressed perpendicular to
the fields. We derive the SPH formalism for the anisotropic heat transport and
solve the corresponding equation with an implicit conjugate gradient scheme. We
discuss several issues of unphysical heat transport in the cases of extreme
ansiotropies or unmagnetized regions and present possible numerical
workarounds. We implement our algorithm into the GADGET code and study its
behaviour in several test cases. In general, we reproduce the analytical
solutions of our idealised test problems, and obtain good results in
cosmological simulations of galaxy cluster formations. Within galaxy clusters,
the anisotropic conduction produces a net heat transport similar to an
isotropic Spitzer conduction model with an efficiency of one per cent. In
contrast to isotropic conduction our new formalism allows small-scale structure
in the temperature distribution to remain stable, because of their decoupling
caused by magnetic field lines. Compared to observations, isotropic conduction
with more than 10 per cent of the Spitzer value leads to an oversmoothed
temperature distribution within clusters, while the results obtained with
anisotropic thermal conduction reproduce the observed temperature fluctuations
well. A proper treatment of heat transport is crucial especially in the
outskirts of clusters and also in high density regions. It's connection to the
local dynamical state of the cluster also might contribute to the observed
bimodal distribution of cool core and non cool core clusters. Our new scheme
significantly advances the modelling of thermal conduction in numerical
simulations and overall gives better results compared to observations.
