Advances in Computational Mathematics

, Volume 45, Issue 2, pp 1067–1103 | Cite as

Numerical modeling of wildland surface fire propagation by evolving surface curves

  • Martin AmbrozEmail author
  • Martin Balažovjech
  • Matej Medl’a
  • Karol Mikula


We introduce a new approach to wildland fire spread modeling. We evolve a 3-D surface curve, which represents the fire perimeter on the topography, as a projection to a horizontal plane. Our mathematical model is based on the empirical laws of the fire spread influenced by the fuel, wind, terrain slope, and shape of the fire perimeter with respect to the topography (geodesic and normal curvatures). To obtain the numerical solution, we discretize the arising intrinsic partial differential equation by a semi-implicit scheme with respect to the curvature term. For the advection term discretization, we use the so-called inflow-implicit/outflow-explicit approach and an implicit upwind technique which guarantee the solvability of the corresponding linear systems by an efficient tridiagonal solver without any time step restriction and also the robustness with respect to singularities. A fast treatment of topological changes (splitting and merging of the curves) is described and shown on examples as well. We show the experimental order of convergence of the numerical scheme, we demonstrate the influence of the fire spread model parameters on a testing and real topography, and we reconstruct a simulated grassland fire as well.


Curve evolution Surface curve Topological changes Wildland fire modeling Geodesic curvature Normal curvature 

Mathematics Subject Classification (2010)

35R01 65M08 53Z05 68U20 


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Funding information

This work was supported by the grants VEGA 1/0608/15 and APVV-15-0522.


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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovak Republic
  2. 2.Algoritmy:SK s.r.o.BratislavaSlovak Republic

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