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

Raimund Bürger,Pep Mulet,Luis Miguel Villada,Daniel Inzunza,Elvis Gavilán

doi:10.3390/math8101674

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.

Highlights

It is the purpose of this contribution to apply a recently-developed efficient numerical method [1]for the solution of a wildland fire model [2] to simulate the propagation of a wildfire in various spatially heterogeneous environments
Scenarios 2.0 to 2.4: Effect of the Variability of Topography In Scenario 2.0 we provide a comparison with a reference solution to show the numerical convergence of S-LIMEX scheme and in Scenarios 2.1 to 2.4 we study the net effect of terrain topography on the dynamics of wildfire propagation
An analysis that is interesting when studying the effects of topography on the source of fire in a forest fire is the elaboration of the so-called risk maps, which we present

Summary

Introduction

It is the purpose of this contribution to apply a recently-developed efficient numerical method [1]for the solution of a wildland fire model [2] to simulate the propagation of a wildfire in various spatially heterogeneous environments. Applications of this model to real-world wildfire scenarios in several specific geographic regions are documented in [4,8,9] It is the basis of the spectral algorithm advanced by San Martin and Torres [10,11], but the particular nature of that numerical method is applicable to a constant diffusion coefficient only. An alternative approach to the description of wildfires through partial differential equations, and their numerical solution by explicit methods is provided in [12,13]. These models and simulators are all based on the principles of combustion theory (see, for instance, [14,15]). Under simplifying assumptions one obtains a fully three-dimensional reaction–diffusion–convection system governed by thermodynamics [16]

Methods

Results

Conclusion