## 3 Results and Discussion  
The accuracy of the described approach, compared against different code results, is shown by considering the classical indicator  
\[Er(x)=\frac{\sum_{j=1}^{m}(p_{x}(\
u_{j})-p_{j})^{2}}{||p_{j}||^{2}} \tag{45}\]  
Several simulation runs were performed, comparing the open literature and personal communication results from three different software:* CHEETAH version 2.0 [27], developed at Lawrence Livermore National Laboratory in the United States of America;
* ZMWNI [20], developed at the Faculty of Advanced Technologies and Chemistry at the Military University of Technology in Poland. * EXPLO5 version 7.1 [5], developed by OZM Research s.r.o. in the Czech Republic. Data points were taken from these programs to compare built-in ideal detonation code fitting routines with the approach described in Section 2.1. For coherence, CJ values obtained by detonation codes were used to constrain the objective function (44). The first numerical test was performed against values from CHEETAH 2.0 [27] software. Isentropic reduced volume-pressure data for PETN (classic energetic compound) at maximum density obtained from CHEETAH 2.0 are presented in Table 1. As previously stated, it can be seen from the results in Table 2 that large deviations in JWL coefficients' estimation are observed, even though the results for the detonation energy \(E_{0}\) are similar. The error \(Er\) comparison shows a better accuracy in the fitting results. The second test was performed against ZMWNI results stored in Table 3, taken from [20] for RDX. The authors employed a similar method for constraining CJ values and \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{\
u}\) & \(\mathbf{p}\) [GPa] \\ \hline
0.767 & 30.788 \\ \hline
1.000 & 13.319 \\ \hline
2.200 & 1.302 \\ \hline
4.100 & 0.345 \\ \hline
6.500 & 0.158 \\ \hline
10.000 & 0.083 \\ \hline
20.000 & 0.032 \\ \hline
40.000 & 0.014 \\ \hline
80.000 & 0.006 \\ \hline
160.000 & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pressure (\(GPa\)) with respect to reduced volume \(\
u\) data from CHEETAH 2.0 software for PETN with \(\rho_{0}=1760\) [Kg of energetic compound/m\({}^{3}\)]. Chapman-Jouguet (CJ) point values from CHEETAH 2.0 are \(v_{\text{CJ}}=0.4356\,\text{cm}^{3}/\text{g}\), \(\rho_{\text{CJ}}=30.788\,\text{GPa}\), \(\gamma_{\text{CJ}}=3.286\). \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **NLLS (This Work)** & **CHEETAH2** \\ \hline \(A\) (GPa) & 1069.7 & 985.014 \\ \hline \(B\) (GPa) & 6.006 & 8.939 \\ \hline \(C\) (GPa) & 1.221 & 1.327 \\ \hline \(R_{1}\) & 4.8410 & 4.719 \\ \hline \(R_{2}\) & 0.9228 & 1.067 \\ \hline \(\omega\) & 0.196 & 0.235 \\ \hline \(E_{0}\) (\(\text{J}/\text{m}^{3}\)) & \(-11.834\) & \(-11.727\) \\ \hline \(Er\) & 0.041 & 0.045 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of JWL parameters obtained using NLLS fitting and CHEETAH 2.0 built-in fitting solver. Even though the results for the coefficients differ, the value of detonation energy \(E_{0}\) differs only by 0.9%. then fit the resulting cost function using a classic Powell method. The results are presented in Table 4, where it is shown again that, despite the differences in JWL parameters, the value of the detonation energy \(E_{0}\) is equal up to five significant digits. Moreover, the error analysis demonstrates the major accuracy of the proposed approach. The last numerical test that was performed against values of isentropic expansion taken from EXPLO5 simulation results available in Table 5. The authors do not describe in detail the fitting procedure implemented in the software: it is generically stated that both pressure and energy data are considered in the minimization process. It is worth noting that, unlike CHEETAH 2.0, ZMWNI, and other codes, EXPLO5 computes more than 50 points for a reduced volume, up to values near \(900v_{0}\). The validity of the JWL EoS up to these values is uncertain when compared to volume ranges considered in the corresponding experimental setups.