## 3 Current Model and Numerical Algorithm
### Model Description  
The surface \(S\) where the fire takes place is defined by the mapping described by the expression (15). The size of this surface 5 and the study time interval \((0,t_{max})\) were selected so that the fire does not reach the boundary in time \(t_{max}\), to assume Dirichlet homogeneous boundary conditions to complete the set of PDEs that governs the PhyFire model. The latest version of these non-dimensional PDEs are as follows:  
\[\partial_{t}e+\beta\mathbf{v}\cdot\
abla e+\alpha u=r+q \text{in }S\times(0,t_{max}), \tag{31}\] \[e\in G(u) \text{in }S\times(0,t_{max}),\] (32) \[\partial_{t}c=-g(u)c \text{in }S\times(0,t_{max}), \tag{33}\]  
where the unknowns are the following variables defined in \(S\times(0,t_{max})\): the dimensionless enthalpy \(e\), the dimensionless solid fuel temperature \(u\), and the solid fuel mass fraction \(c\). The relationship between these dimensionless variables and the corresponding physical quantities is given in Table 1.  
The physical quantities that appear in Table 1 are the enthalpy \(E\), the solid fuel temperature \(T\), and the fuel load \(M\). Their relationship depends on the reference temperature \(T_{\infty}\), the heat capacity of the solid fuel \(C\), and the maximum solid fuel load \(M_{0}\). The reference temperature \(T_{\infty}\) is related to the ambient temperature, measured far enough from the fire front to ensure that \(T\geq T_{\infty}\), whereby \(u\geq 0\). The heat capacity of the solid fuel \(C\) and maximum solid fuel load \(M_{0}\) both depend on the fuel type. Table 2 lists all the variables that depend on the fuel type. The PhyFire model also depends on the three model parameters listed in Table 3. These three parameters are related to the heat transfer terms in Equation (31): \(H\) in the natural convection term \(\alpha u\), \(\beta\) in the convective term \(\beta\mathbf{v}\cdot\
abla e\), and \(a\) in the radiation term \(r\) \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Physical Variable** & **Symbol** & **Units** & **Dimensionless Variable** \\ \hline Enthalpy & \(E\) & \(J\,\text{m}^{-2}\) & \(e=E/MCT_{\infty}\) \\ Solid fuel temperature & \(T\) & K & \(u=(T-T_{\infty})/T_{\infty}\) \\ Solid fuel load & \(M\) & \(\text{kg}\,\text{m}^{-2}\) & \(c=M/M_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless unknowns and related physical quantities. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Input Variable** & **Symbol** & **Units** \\ \hline Heat capacity & \(C\) & \(J\,\text{K}^{-1}\,\text{kg}^{-1}\) \\ Maximum initial fuel load & \(M_{0}\) & \(\text{kg}\,\text{m}^{-2}\) \\ Fuel moisture content & \(M_{0}\) & \(\text{kg}\,\text{water}/\text{kg}\,\text{fuel}\) \\ Maximum flame temperature & \(T_{f,max}\) & K \\ Pyrolysis temperature & \(T_{p}\) & K \\ Combustion half-life & \(t_{1/2}\) & s \\ Flame length independent factor & \(F_{H}\) & m \\ Flame length wind correction factor & \(F_{v}\) & \(\text{m}^{1/2}\,\text{s}^{1/2}\) \\ Flame length slope correction factor & \(F_{s}\) & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuel-type-dependent input variables. We completed the problem with the following initial conditions:  
\[u(x,y,0)=u_{0}(x,y) \text{in }S, \tag{34}\] \[c(x,y,0)=c_{0}(x,y) \text{in }S. \tag{35}\]  
which represent, respectively, the focus of the fire, or eventually an intermediate perimeter if it is decided to restart the simulation from an instant later than the initial one, and the initial distribution of the fuel, including any possible firebreaks. We should point out that the data for topography (function \(h(x,y)\) defining \(S\)), solid fuel distribution (initial value of \(M\)), and fuel type distribution (to spatially determine the fuel-type-dependent input variables) were obtained from the cartography generated particularly for each case. This process was automated by developing the necessary cartographic database [17] and the online GIS interface [18].