single ionization rate coefficients that we use and compared the results to the CIE results derived from the @Mazzotta:AAS:1998 rate coefficients. They also found order of magnitude differences at low temperatures when comparing these different data sources. This discrepancy demonstrates the importance of having reliable ionization and recombination rate coefficients.

Direct Calculations {#subsec:numerical}
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For systems in which $T$ or $n_{\mathrm{e}}$ evolve in time, it is necessary to numerically solve Equation (\[eq:dydt\]) to find the CSD at each time step. One challenge in doing this is that Equation (\[eq:dydt\]) represents a “stiff” system of ordinary differential equations. That is, the coefficients on the right hand side of the equation can vary by orders of magnitude. Standard numerical methods have been developed for dealing with such stiff equations [e.g., @Press:Book]. In order to ensure accuracy, we have used an adaptive time step for the integration. Conditions for adapting the timestep have been given by @MacNeice:SolPhys:1984 and used more recently by @Bradshaw:AA:2009. These conditions define values $\epsilon_d$ and $\epsilon_{r}$ and require that the time step be small enough that, for all $i$ $$|y_{i}(t+\Delta t)-y_{i}| \leq \epsilon_{d},
\label{eq:cond_epsilond}$$ and $$|\log [y_{i}(t+\Delta t)] - \log [y_{i}(t)]| \leq \epsilon_{r}. 
\label{eq:cond_epsilonr}$$ @MacNeice:SolPhys:1984 and @Bradshaw:AA:2009 have found 