With the inclusion of the diffusion term in the energy equation we need to ensure that the difference equations remain stable. One technique employed in finite difference methods is to use a linear stability analysis on the difference equations. The same method can be employed with SPH equations, albeit with some simplifying approximations. The following method has been employed on the full set of equations to ensure stability [6,2,9], but here we will concentrate on the diffusion term only.
The stability criterion for the diffusion term in the energy equation can be calculated independently of the equations of motion as it's stability is determined on the thermal timescale not the dynamic timescale.
Writing the energy equation in terms of the specific internal energy U, and assuming a perfect gas, we have
For this analysis we have ignored the adiabatic work term.
We will assume a uniform distribution of particles in one dimension, with a
separation 
, so the density is constant and given by
. We define the new variables 
,
, and 
, which are the perturbations from a uniform
value, so that
Starting with the energy equation with just the diffusion term, given by
the first order perturbation equation, using equations (26), is
If we assume that the perturbation can be written in the form
then equation (28) becomes, on substitution of 
,
For sufficiently small h we can replace the summation above by integrals, equation (29) becomes
where 
and depends on the interpolation kernel used.
By applying the difference scheme being used to equation (31)
an estimate of the stability criterion required can be obtained.
For the predictor-corrector or modified euler scheme employed by a number
of authors [7] the stability criterion is
If a gaussian kernel is used then 
and the criterion above is
the same criterion for an explicit finite difference approximation.
If the B-Spline kernel 
is used the term I is considerably more
complicated but the most restrictive stability criterion is 
.
