\subsection{Stability and convergence of the Enskog solutions}

Before studying the physical predictions acquired from the Enskog solutions a convergence and stability analysis was conducted. It has previously been shown that using the analytic hard sphere solutions for the collision integrals the solutions converge at around approximation order $\approx$ 4-5, and that numerical instability becomes significant at orders $> 7$.\cite{prosjektoppgave} The notable result from the previous analysis is the dramatic increase in the condition number of the matrix of Equation \eqref{eq:kin_d_matrix_form} with increasing order of approximation, resulting in floating point error propagating to large errors in the matrix inversion.

A simple convergence analysis reveals that the same is true when employing a Mie potential as when using the HS-potential, the solutions destabilise at approximation order $\approx 7-8$. Given the previously obtained results regarding the nature of this instability, it was not investigated further.\cite{prosjektoppgave} An interesting discovery however, was that the diffusion factors $d_{-1}$, $d_0$ and $d_1$ converged far more quickly for a Mie potential than for a HS model. In most cases the changes were on the order of 1 \% from the first approximation to the converged value at the 3rd to 5th approximation. 

The reason for this more rapid convergence for the Mie potential was investigated, but no clear explanation was found. However, it was noted that if one component was in trace concentrations ($< 1\ $ \textperthousand) the convergence took longer, requiring up to the 6th or 7th approximation before converging. As previously found,\cite{prosjektoppgave} having one component in trace concentration stabilises the numerics, such that one can go to higher order approximations before the solutions become unstable.

Following these observations, the 3rd Enskog approximation was chosen as the default for all computations. This order of approximation is used unless a different order is explicitly stated.