Stable anisotropic heat conduction in smoothed particle hydrodynamics

We investigate how to simulate anisotropic heat conduction in a stable manner in Smoothed Particle Hydrodynamics. We show that the requirement for stability is that entropy must increase. From this, we deduce that methods involving direct second derivatives in SPH are unstable, as found by previous authors. We show that the only stable method is to use two first derivatives with alternating differenced and symmetric SPH derivative operators, with the caveat, that one may need to apply smoothing or use an artificial conductivity term if the initial temperature jump is discontinuous. Furthermore, we find that with two first derivatives the stable timestep can be 3--8 times larger even for isotropic diffusion.