## 2 Previous Models
### First Model: Turbulent Diffusivity  
The first simplified physical model for wildfire spread simulation proposed was a two-dimensional one-phase model considering turbulent flow, vertical heat lost, and a convective term representing the effect of wind [22]. The original equations are  
\[\rho C\left(\frac{\partial}{\partial t}T+\mathbf{v}\cdot\
abla T\right)- \
abla(K\
abla T)+H(T)(T-T_{\infty})=Q\,M\,A\,e^{-\frac{E_{A}}{RT}}, \tag{3}\]  
\[\frac{\partial}{\partial t}M=-M\,A\,e^{-\frac{E_{A}}{RT}}, \tag{4}\]  
where \(T\) is the average value of temperature and \(M\) is the solid fuel load. \(K\) is the turbulent diffusivity. \(H(T)=\overline{H}(T-T_{\infty})^{1/3}\) is the vertical convection heat transfer. \(T_{\infty}\) is the ambient temperature. The reaction rate is given by the Arrhenius law. \(R\) is the universal gas constant. \(E_{A}\) represents the activation energy. \(A\) is the pre-exponential or frequency factor of the reaction. We assumed that the activation energy for the gaseous phase is much lower than the one of the solid phase in order to propose a simplified one-phase chemical reaction. \(Q\) is the global heat production in the reaction.  
The PDEs to be solved are a non-dimensional version of Equations (3) and (4), based on the following change of variables: the dimensionless temperature is \(u=(T-T_{\infty})/(\overline{T}-T_{\infty})\), where \(\overline{T}\) is a temperature up to which generalized combustion takes place; the dimension less solid fuel is \(c=M/M_{0}\), where \(M_{0}\) the initial solid fuel load. Dimensionless time and space variables were also introduced, although, for the sake of simplicity, we kept the same notation. The non-dimensional equations are  
\[\frac{\partial}{\partial t}u+\mathbf{w}\cdot\
abla u-\
abla(\kappa\
abla u)+h(u )u=q\,c\,s(u), \tag{5}\]  
\[\frac{\partial}{\partial t}c=-c\,s(u), \tag{6}\]  
where \(\mathbf{w}\) is the dimensionless wind velocity field, \(\kappa\) is related to the turbulent diffusivity, \(h(u)\) is the normalized vertical convection coefficient, \(q\) depends on the heat released, and \(s(u)\) is written in terms of the previous exponential expression.  
The non-dimensional equations were completed with the initial and boundary conditions. We propose the Dirichlet boundary conditions assuming the domain is big enough to ensure that the fire does not arrive at the boundary. The initial conditions represent the initial fuel load and fire source.  
The first numerical method proposed was an implicit and upwind finite difference scheme for the energy Equation (5), assuming that the wind field velocity and diffusivity are constant along the domain. The matrix of the linear system obtained in each time step is block tridiagonal and strictly diagonally dominant, so the blockwise Gauss-Seidel method is convergent. The global Lipschitz condition that the source term satisfies ensures the stability of the scheme. An implicit Euler method was the scheme proposed for the fuel Equation (6).  
The second numerical method proposed for the partial differential problem given by Equations (5) and (6) was an Adaptive Finite-Element Method (AFEM). The idea of using an adaptive method aims to reduce the operational cost by adapting the mesh to the numerical solution where more precision is necessary, the fire front, as this takes up a small part of the whole domain. This method combines refinement and derefinement techniques to generate in each time step a sequence of nested meshes, allowing an easy application of multigrid acceleration techniques. The design of this first model and its numerical resolution already pursued the aim of simulating the fire spread in times much shorter than real-time, through a simplified model and efficient numerical methods.