# Stable anisotropic heat conduction in smoothed particle hydrodynamics

## TL;DR

This paper presents a stable method for simulating anisotropic heat conduction in Smoothed Particle Hydrodynamics, emphasizing entropy increase and proposing a two-first-derivative approach for improved stability and larger timesteps.

## Contribution

It introduces a novel stable SPH method using two first derivatives with alternating operators for anisotropic heat conduction, avoiding instability of second derivative methods.

## Key findings

- Stable method requires entropy increase.
- Two first derivatives with alternating operators are stable.
- Larger timesteps are possible with the proposed method.

## Abstract

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.