A simple one dimensional test of equation (8) is calculating the heat flow in a bar. The equation to be solved is
with boundary conditions 
.
Figure 1: The number density of the distribution 
, with 
and 51 points. Lines have been added for clarity only.
The interpolation points have to be distributed in the solution domain
. A simple distribution is to have them equally spaced, but this
will not give us any indication how the SPH approximation is affected by
the particle distribution. A more demanding distribution is to place them
by the criterion 
, where `Mod' means only the fractional
part is retained, and i is the particle number.
The number density for this distribution using 50 particles is shown in
figure (1).
As the SPH equation we are testing requires a boundary condition, a simple modification to the SPH equation is to add the term
which produces the correct behavor at the boundaries.
Using the particle distribution of figure (1), the steady state solution of equation (12) is shown in figure (2). The solution labeled direct calculates the energy generation term directly on the particles. The solution labeled smoothed, calculates the energy generation term directly on the particles then smooths the result using the following equation,
Figure 2: The steady state solution for the one dimensional test case.
This form of the interpolated energy generation term uses the derivative of the kernel as the interpolation function in the same way as the second derivative calculation. This form was chosen for consistency and ease of calculation. The energy generation can be calculated in the same loop as the diffusion term.
From figure (2) we can see that extreme particle distributions have very little effect on the solution. The double summation method of equations (3) and (4) was also used to solve equation (12). The solution developed instabilities quickly which grew with time, an indication that the double summation method is sensitive to particle dissorder. The double summation method was also tried with equi-spaced particles, here it performed better but instabilities grew at the boundaries (where errors are the highest) and propogated into the solution quickly.
The comparison of the direct calculation of the energy generation and the interpolated energy generation demonstrates the need for consistency in converting equations into an SPH form.
