## 2 Materials and Methods
### JWL Fitting Methods  
In the context of the CJ detonation theory [21], a complete JWL parameter set \(A,B,R_{1},R_{2},\omega,E_{0}\) implicitly defines the detonation velocity \(D_{\text{CJ}}\), the detonation pressure \(p_{\text{CJ}}\), the relative volume at the CJ-condition \(v_{\text{CJ}}\), and and the JWL-parameter \(C\). It is thereby common to provide \(E_{0}\) instead of \(C\) because of the practical relevance of the usable detonation energy. Figure 2: The plot shows how to compute the detonation energy \(E_{0}\) from the JWL curve and the Rayleigh line. The areas under the isentropic curve fitted by the JWL product EoS and the Rayleigh line, in fact, represent the work in the \(pv\) plane. The energy of compression given by the shock wave passage \(E_{c}\) is represented by the triangular area under the Rayleigh line (blue); the chemical energy released is represented by the dashed red area under the JWL curve; \(E_{0}\), hence, is the difference (green) between the 2 areas. One of the primary methods for determining these constants is the cylinder test [22]: detailed descriptions of these experiment-based methods can be found in [7; 23]. Over the years, several methods have been employed to fit JWL parameters [24; 25]. The form of the JWL equation does not allow for the use of \"brute force\" fitting methods, since the coefficients are not independent. There is no optimal set of coefficients; an error in one of them can be easily compensated for by the \"error\" of others. Additional conditions can be applied by using the CJ theory. One possibility is the one introduced by [PERSON] in [9] that employs a global optimization method based on particle swarm to obtain a global minimum, given that the parameters are not independent. A different approach comes from CTEv [26], in which the optimization is carried out through the minimization of the error function between the experimental points given by energy \(E_{d}\) and reduced volumes \(\
u\) and the expression for detonation energy  
\[E_{d}=E_{0}-\frac{A\mathrm{e}^{-R_{1}v_{C}}}{R_{1}}-\frac{A\mathrm{e}^{-R_{2} v_{C}}}{R_{2}}-\frac{C}{\omega\upsilon_{CJ}\omega},\]  
where coefficients \(A,B,\) and \(C\) are fixed and computed at every step of the optimization loop by solving the linear system that enforces the constraints given from the knowledge of the CJ point  
\[A\mathrm{e}^{-R_{1}v_{C}}+B\mathrm{e}^{-R_{2}v_{C}}+\frac{C}{ \upsilon_{CJ}^{\omega+1}}=p_{CJ}\] CJ pressure \[-\frac{A\mathrm{e}^{-R_{1}v_{C}}}{R_{1}}-\frac{B\mathrm{e}^{-R_{2} v_{C}}}{R_{2}}-\frac{C}{\omega\upsilon_{CJ}^{\omega}}= -E_{0}-\frac{p_{CJ}\big{(}1-\upsilon_{CJ}\big{)}}{2}\] CJ energy \[AR_{1}\mathrm{e}^{-R_{1}v_{C}}+BR_{2}\mathrm{e}^{-R_{2}v_{C}}+ \frac{C(\omega+1)}{\upsilon_{CJ}^{\omega+2}}= \frac{p_{CJ}}{\big{(}1-\upsilon_{CJ}\big{)}}\] CJ tangency  
Alternatively, the JWL constants can be derived from the expansion isentrope pressure-volume data \((\
u_{j},p_{j})\) obtained through thermochemical calculations: after determining the detonation parameters of the explosive material (the CJ point), it is possible to calculate the expansion isentrope of the product mixture at selected points on the \((\
u,p)\) plane, as described in [16]. These points are obtained by constraining the condition of constant entropy; \(S=\mathrm{const}\). This method is employed in several other thermochemical codes aimed at energetic material simulations, such as CHEETAH 2.0 [27], EXPLO5 [5], and ZMWNI [20]. The isentrope data obtained by the thermochemical routines can then be used to determine the parameters of the JWL equation, minimizing the residual \(r_{j}(x)\). Generally, in this second approach, it is possible to exclude \(E_{0}\) from the parameter set that needs to be determined by the minimization procedure and instead provide \(C,D,p_{CJ}\) or \(r_{CJ}\) to implicitly define \(E_{0}\) from (32). Therefore, one must take care to maintain consistency while providing redundant implicit parameters for the JWL that are just a necessity of the underlying CJ conditions.