## 3 Methodology
### Combustion Fitting Approach  
The Newton-Raphson method applied for a single variable function is briefly introduced. If the initial estimate of the root is \(x_{n}\), a tangent can be extended from the point \([x_{n},f(x_{n})]\). The point where this tangent crosses the x-axis represents an improved estimate of the root. The first derivative \(f^{\prime}(x_{n})\) is equivalent to the slope of the tangent and can be derived on the basis of a geometrical interpretation (Figure 6) and an iteration scheme is constructed:  
\[f^{\prime}(x_{n})=\frac{f(x_{n})-0}{x_{n}-x_{n+1}} \tag{2}\]  
Then  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})} \tag{3}\]  
While \(\mid x_{n+1}-x_{n+1}\mid\leq\)_tolerance_, the \(x_{n+1}\) can be considered to be the root of function \(f(x)\).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Seiliger Stage** & **Seiliger Definition** & **Parameter Definition** & **Seiliger Parameters** \\ \hline
1–2 & \(\frac{P_{2}}{P_{1}}=r_{c}^{t_{comp}}\) & \(\frac{V_{1}}{V_{2}}=r_{c}\) & \(r_{c}\), \(n_{comp}\) \\
2–3 & \(\frac{V_{2}}{V_{1}}=1\) & \(\frac{P_{2}}{P_{2}}=a\) & \(a\) \\
3–4 & \(\frac{P_{2}}{P_{3}}=1\) & \(\frac{V_{3}}{V_{4}}=b\) & \(b\) \\
4–5 & \(\frac{T_{2}}{P_{4}}=1\) & \(\frac{V_{3}}{V_{4}}=c\) & \(c\) \\
5–6 (5–6’) & \(\frac{P_{5}}{P_{6}}=r_{e}^{n_{exp}}\) & \(\frac{V_{3}}{V_{5}}=r_{e}\) & \(r_{c}\), \(n_{exp}\) \\ \hline \end{tabular}
\end{table}
Table 2: Seiliger process definition and parameters.  
Figure 5: Flow chart of the overall simulation procedure to calculate Seiliger parameters.  
Furthermore, the Newton-Raphson method can be used to acquire the solution for a multi-variable function **f(x)**. If the zero was considered to occur at \(\mathbf{x}=\mathbf{x}^{\star}\), where \(\mathbf{x}^{\star}\) is a vector, then the Taylor series for multi-variable function is applied. If \(\mathbf{x_{n}}\) is assumed to be the current estimation of the functions and \(\mathbf{x_{n+1}}=\mathbf{x_{n}}+\mathbf{h}\), due to the demand that \(\mathbf{f(x^{\star})=0}\), the Taylor series is acquired:  
\[\mathbf{f(x)}|_{\mathbf{x}\mathbf{x_{n+1}}}=\mathbf{f(x_{n}+h)}=\mathbf{f(x_{ n})}+\left.\
abla\mathbf{f}|_{\mathbf{x_{n}}}\cdot\mathbf{h}+\mathbf{o}\left(| \mathbf{h}|^{2}\right)=\mathbf{0}\right. \tag{4}\]  
If \(\mathbf{o}\left(|\mathbf{h}|^{2}\right)\) is assumed small enough to neglect, Equation (4) becomes:  
\[\mathbf{f(x_{n})}=-\left.\
abla\mathbf{f}\right|_{\mathbf{x_{n}}}\cdot\mathbf{h} \tag{5}\]  
When \(-\left.\
abla\mathbf{f}\right|_{\mathbf{x_{g}}}\) is replaced by matrix \(\mathbf{A}\):  
\[\mathbf{h}=-\mathbf{A}^{-1}\cdot\mathbf{f(x_{n})} \tag{6}\]  
\[\mathbf{A}=\left[\begin{array}{cccc}\frac{\partial f_{1}}{\partial x_{1}}& \frac{\partial f_{1}}{\partial x_{2}}&\cdots&\frac{\partial f_{1}}{\partial x _{n}}\\ \frac{\partial f_{2}}{\partial x_{1}}&\frac{\partial f_{2}}{\partial x_{2}}& \cdots&\frac{\partial f_{2}}{\partial x_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial f_{n}}{\partial x_{1}}&\frac{\partial f_{n}}{\partial x_{2}}& \cdots&\frac{\partial f_{n}}{\partial x_{n}}\\ \end{array}\right]\text{ and }\mathbf{f(x)=\left[\begin{array}{c}f_{1}(x_{1},x_{2}, \cdots x_{n})\\ f_{2}(x_{1},x_{2},\cdots x_{n})\\ \vdots\\ f_{n}(x_{1},x_{2},\cdots x_{n})\\ \end{array}\right] \tag{7}\]  
Finally, the next estimate of \(\mathbf{x_{n+1}}\) is obtained and used as the initial value of the next iteration and the iteration is terminated when the stop criteria are met.