Lagrangean Method with Topological Changes for Numerical Modelling of Forest Fire Propagation

We introduce a mathematical model and new Lagrangean numerical algorithm for modelling of a wind-driven forest fire front propagation. The model is based on evolution of plane curve (representing the fire front) in the outer normal direction by a speed given by the properties of a fuel bed which is scaled exponentially by a wind speed projected onto the normal to the front. The influence of the front shape on the speed of propagation is modelled by adding the curvature regularization to the normal velocity. For numerical modelling we use so-called Lagrangean approach where the crucial point is an asymptotically uniform tangential redistribution of grid points which prevents the moving front from forming spurious crossovers and swalow tails. Moreover, thanks to the uniform tangential redistribution and our new idea of computing distance function in a narrow tube along discrete curve grid points, we detect and solve in O(n) complexity the topological changes in moving front. Such approach makes our Lagrangean method highly efficient and represents significant improvement of the existing numerical models for the forest fire propagation and, in general, it represent new fast and stable method for solving free boundary problems modelled by moving fronts with possible topological changes.