Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

Abstract

The propagation of a forest fire can be described by a convection–diffusion–reaction problem in two spatial dimensions, where the unknowns are the local temperature and the portion of fuel consumed as functions of spatial position and time. This model can be solved numerically in an efficient way by a linearly implicit–explicit (IMEX) method to discretize the convection and nonlinear diffusion terms combined with a Strang-type operator splitting to handle the reaction term. This method is applied to several variants of the model with variable, nonlinear diffusion functions, where it turns out that increasing diffusivity (with respect to a given base case) significantly enlarges the portion of fuel burnt within a given time while choosing an equivalent constant diffusivity or a degenerate one produces comparable results for that quantity. In addition, the effect of spatial heterogeneity as described by a variable topography is studied. The variability of topography influences the local velocity and direction of wind. It is demonstrated how this variability affects the direction and speed of propagation of the wildfire and the location and size of the area of fuel consumed. The possibility to solve the base model efficiently is utilized for the computation of so-called risk maps. Here the risk associated with a given position in a sub-area of the computational domain is quantified by the rapidity of consumption of a given amount of fuel by a fire starting in that position. As a result, we obtain that, in comparison with the planar case and under the same wind conditions, the model predicts a higher risk for those areas where both the variability of topography (as expressed by the gradient of its height function) and the wind velocity are influential. In general, numerical simulations show that in all cases the risk map with for a non-planar topography includes areas with a reduced risk as well as such with an enhanced risk as compared to the planar case.

Modeling Fire Hazards Induced by Volcanic Eruptions: The Case of Stromboli (Italy)

“…The more comprehensive full-physics-based models [16,17] are also computationally expensive (e.g., with representation of vegetation on the mesoscale as a porous medium and/or using balancing of the energy/heat equation), while others prefer a simplified physical model to achieve faster computations, for example solving 2D reaction/diffusion equations [15,18], or using even less physics, such as in estimate-based models using graphs [19], cellular automata [20][21][22][23] or envelope-based approaches [24]. Among the numerical models for simulating the propagation of wildfires, ignition mechanisms that utilise various input parameters are taken into consideration, including different types of trees, canopies, and the influence of wind [25], moisture content, radiant capacity [26], and/or topography [27]. The integration of these models into GIS environments allows for both soil classification (zoning for susceptibility, hazard, and exposure estimation), considering the presence of vegetation and/or infrastructures [10,22], and the possibility of mapping burnt areas using satellite data [28].…”