We have presented here an alternate form for the solution of second order derivatives. To calculate the second derivative this method only needs to interpolate from the particles once. A standard SPH approximation needing to interpolate from the particles twice was found to be unstable for a simple one dimensional problem. The new method was found to be surprisingly stable.
As errors in the SPH approximation increases in the vicinity of a free surface, no reliance can be made on the interpolation method implicitly producing the correct radiative boundary condition. For this reason a radiative surface condition was developed that works well for equilibrium figures, but would be inadequate for problems that contained density discontinuities.