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

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^{2}MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción 4030000, Chile

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^{3}

^{4}

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^{2}MA, Universidad de Concepción, Casilla 160-C, Concepció n 4030000, Chile

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## Abstract

**:**

## 1. Introduction

## 2. Summary of the Mathematical Model

## 3. Numerical Method

## 4. Numerical Results

#### 4.1. Preliminaries

#### 4.2. Scenarios 1.1 to 1.4: Numerical Experiments with Various Diffusion Coefficients

#### 4.3. Scenarios 2.0 to 2.4: Effect of the Variability of Topography

#### 4.4. Scenarios 3.1 to 3.4: Risk Maps

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

List of symbols: | |

Latin symbols: | |

A | pre-exponential factor [s^{−1}] |

C | specific heat [kJ/(kg K)] |

$c\left(u\right)$ | phase change function [-] |

$\mathcal{C}\left(\mathit{u}\right)$ | discretization of convective term [-] |

$\mathcal{D}\left(\mathit{u}\right)$ | discretization of diffusive term [-] |

${E}_{A}$ | reaction energy [ kcal/mol] |

${\widehat{f}}_{\mathit{i}+\frac{1}{2}{\mathit{e}}_{l}}$ | numerical flux in the l-coordinate [-] |

$f(u,v,\mathit{x})$ | reactive term for u [-] |

$g(u,v)$ | reactive term for v [-] |

$\mathit{i}=(i,j)$ | index vector [-] |

k | conductivity coefficient [W/(mK)] |

${K}_{\mathit{i}}$ | approximate value $K\left({\mathit{u}}_{\mathit{i}}\right)$ [-] |

$K\left(u\right)$ | diffusion coefficient function [-] |

L | length of domain x or y coordinate [-] |

${l}_{0}$ | length scale [m] |

M | number points x or y coordinate [-] |

${q}_{\mathrm{react}}$ | reaction heat [-] |

r | Arrhenius expression [-] |

R | Universal gas constant [cal/(K mol)] |

$\mathcal{T}$ | topography term [-] |

t | time variable [-] |

${t}^{n}$ | time discretization [-] |

${t}_{0}$ | time scale [s] |

u | average temperature [-] |

${u}_{\mathrm{pc}}$ | phase change temperature [-] |

${u}_{max}$ | maximum temperature [-] |

U | absolute temperature [K] |

${U}_{\mathrm{ref}}$ | ambient temperature [K] |

${\mathit{u}}^{n}$ | vector of components ${\mathit{u}}_{\mathit{i}}^{n}$ |

${u}_{\mathit{i}}^{n}$ | approximate value $u({\mathit{x}}_{\mathit{i}},{t}^{n})$ [-] |

v | mass fraction of solid fuel [-] |

${v}^{n}$, | vector of components ${v}_{\mathit{i}}^{n}$ [-] |

${v}_{\mathit{i}}^{n}$ | approximate value $v({\mathit{x}}_{\mathit{i}},{t}^{n})$ [-] |

w | vector field [-] |

${w}_{0}$ | wind speed [-] |

$\mathit{x}=(x,y)$ | spatial variables [-] |

${\mathit{x}}_{\mathit{i}}=({x}_{i},{y}_{j})$ | nodes in the Cartesian grid [-] |

Greek symbols: | |

$\alpha $ | natural convection coefficient [-] |

$\delta $ | length of optical path for radiation [-] |

$\Delta x$, $\Delta t$ | space and time discretizations [-] |

$\epsilon $ | inverse of the activation energy [-] |

$\kappa $ | inverse of conductivity coefficient [-] |

$\rho $ | density of the fuel [kg/m^{3}] |

$\sigma $ | Stefan-Boltzmann constant |

${\phi}_{\Delta t}(\mathit{u},v)$ | solution operator of (12) [-] |

${\psi}_{\Delta t}(\mathit{u},v)$ | solution operator defined in (11) [-] |

$\mathsf{\Omega}\times (0,\infty )$ | domain [-] |

Abbreviations: | |

CFL | Courant–Friedrichs–Lewy |

CPU | central processing unit |

DIRK | diagonally implicit Runge–Kutta |

ERK | explicit Runge–Kutta |

H-LDIRK3(2,2,2) | acronym of particular IMEX-RK scheme |

IMEX | implicit–explicit |

IMEX-RK | implicit–explicit Runge–Kutta |

LI-IMEX | linearly implicit–explicit |

NI-IMEX | nonlinearly implicit–explicit |

ODE | ordinary differential equation |

PDE | partial differential equation |

RK | Runge–Kutta |

S-LIMEX | Strang-type splitting scheme |

SSP | strong stability-preserving |

WENO | weighted essentially non-oscillatory |

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**Figure 2.**Scenarios 1.1 to 1.4: three-dimensional plots of simulated temperature and two-dimensional plots of simulated burnt fuel, at simulated time $T=12$. The red and blue parts correspond to areas with ${v}_{0}=0.9$ and ${v}_{0}=0$, respectively. The small white square indicates the location of the initial fire focus. The percentages represent the portion of burnt fuel of the total fuel initially available in $\mathsf{\Omega}$.

**Figure 3.**Topographies for $\mathcal{T}(x,y)$ given by (15) with ${h}_{1}={h}_{2}=60$ and (

**left**) ${x}_{1}=0$, ${y}_{1}=150$, ${\gamma}_{1}=10,000$, ${x}_{2}=150$, ${y}_{2}=150$, ${\gamma}_{2}=300$ (topography ${\mathcal{T}}_{1}$ for Scenario 2.2), (

**middle**) ${x}_{1}=150$, ${y}_{1}=50$, ${\gamma}_{1}=1000$, ${x}_{2}=50$, ${y}_{2}=150$, ${\gamma}_{2}=1000$ (topography ${\mathcal{T}}_{2}$ for Scenario 2.3), (

**right**) ${x}_{1}=150$, ${y}_{1}=75$, ${\gamma}_{1}=3000$, ${x}_{2}=50$, ${y}_{2}=150$, ${\gamma}_{2}=2000$ (topography ${\mathcal{T}}_{3}$ for Scenario 2.4), shown in each case as three-dimensional plots (

**top**) and as contour maps (

**bottom**).

**Figure 4.**Scenario 2.0: accuracy test. Numerical approximations at simulated time $T=0.2$ and $T=0.4$ obtained with the S-LIMEX scheme with $\Delta x=80/M$ and $M=160$ and $M=640$ (which correspond to the reference solution for computation of approximate errors in Table 1).

**Figure 5.**Scenarios 2.1 and 2.2: simulation of the propagation of a forest fire on a flat domain (first and second row) and on a domain with topography ${\mathcal{T}}_{1}$ (third and fourth row), at indicated simulated times. Here and in Figure 6, the first and third row show temperature and the second and fourth row show fuel, and small white square indicates the location of the initial wildfire.

**Figure 6.**Scenarios 2.3 and 2.4: simulation of the propagation of a forest fire on domains with topographies ${\mathcal{T}}_{2}$ (first and second row) and ${\mathcal{T}}_{3}$ (third and fourth row).

**Figure 7.**Scenario 3.1: principle of construction of a risk map for a flat domain. Here and in Scenarios 3.2 to 3.4, the propagation of a wildfire is simulated starting from an array of $5\times 5=25$ initial fire foci, marked by small white squares. For each initial position the simulation is stopped when $5\%$ of the initially available fuel is burnt. The corresponding times ${T}_{\mathrm{risk}}$ are mapped in Figure 8. (For the present flat case, the shapes of the temperature and fuel distributions are the always the same in relation to the initial fire; instead of displaying $2\times 25$ identical results, we only show the ‘corner’ cases of the $5\times 5$ initial positions).

**Figure 8.**Scenarios 3.1 to 3.4: risk maps for the subdomain $[60,110]\times [60,110]$ under the assumption of a flat domain (Scenario 3.1, top left) and domains with topography ${\mathcal{T}}_{1}$, ${\mathcal{T}}_{2}$, and ${\mathcal{T}}_{3}$ (Scenarios 3.2 to 3.4, top right, bottom left, and bottom right), visualizing for each patch of size $10\times 10$ the final time ${T}_{\mathrm{risk}}$. A low value of ${T}_{\mathrm{risk}}$ implies a high risk.

**Figure 9.**Scenario 3.2: simulation of temperature of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{1}$. Here and in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, for each initial position where the simulation is stopped, and the numerical solutions for u and v are portrayed, when $5\%$ of the initially available fuel is burnt. The corresponding simulated time determines ${T}_{\mathrm{risk}}$ for each initial position.

**Figure 10.**Scenario 3.2: simulation of fuel of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{1}$.

**Figure 11.**Scenario 3.3: simulation of temperature of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{2}$.

**Figure 12.**Scenario 3.3: simulation of fuel of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{2}$.

**Figure 13.**Scenario 3.4: simulation of temperature of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{3}$.

**Figure 14.**Scenario 3.4: simulation of fuel of a wildfire from $5\times 5=25$ initial fire foci, marked by small white squares, on a domain with topography ${\mathcal{T}}_{3}$.

**Table 1.**Scenario 2.0: Approximate ${L}^{1}$ errors ${e}_{M}\left(u\right)$ and ${e}_{M}\left(v\right)$ for Strang LIMEX method with $\Delta x=80/M$. Reference solution computed with $M=640$.

$\mathit{T}=0.2$ | $\mathit{T}=0.3$ | $\mathit{T}=0.4$ | ||||
---|---|---|---|---|---|---|

$M$ | ${e}_{M}\left(u\right)$ | ${e}_{M}\left(v\right)$ | ${e}_{M}\left(u\right)$ | ${e}_{M}\left(v\right)$ | ${e}_{M}\left(u\right)$ | ${e}_{M}\left(v\right)$ |

40 | 2.2 | 3.7 $\times {10}^{-2}$ | 3.2 | 4.5$\times {10}^{-2}$ | 4.9 | 6.4$\times {10}^{-2}$ |

80 | 1.2 | 1.6 $\times {10}^{-2}$ | 2.0 | 2.3 $\times {10}^{-2}$ | 3.3 | 3.7$\times {10}^{-2}$ |

160 | 0.7 | 1.0 $\times {10}^{-2}$ | 1.2 | 1.5 $\times {10}^{-2}$ | 2.0 | 2.4$\times {10}^{-2}$ |

320 | 0.4 | 5.7 $\times {10}^{-3}$ | 0.6 | 8.9 $\times {10}^{-3}$ | 1.1 | 1.4$\times {10}^{-2}$ |

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## Share and Cite

**MDPI and ACS Style**

Bürger, R.; Gavilán, E.; Inzunza, D.; Mulet, P.; Villada, L.M. Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments. *Mathematics* **2020**, *8*, 1674.
https://doi.org/10.3390/math8101674

**AMA Style**

Bürger R, Gavilán E, Inzunza D, Mulet P, Villada LM. Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments. *Mathematics*. 2020; 8(10):1674.
https://doi.org/10.3390/math8101674

**Chicago/Turabian Style**

Bürger, Raimund, Elvis Gavilán, Daniel Inzunza, Pep Mulet, and Luis Miguel Villada. 2020. "Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments" *Mathematics* 8, no. 10: 1674.
https://doi.org/10.3390/math8101674