This paper presents an explicit finite-difference method for nonlinear partial differential equation appearing as a transformed Black-Scholes equation for American put option under logarithmic front fixing transformation. Numerical analysis of the method is provided. The method preserves positivity and monotonicity of the numerical solution. Consistency and stability properties of the scheme are studied. Explicit calculations avoid iterative algorithms for solving nonlinear systems. Theoretical results are confirmed by numerical experiments. Comparison with other approaches shows that the proposed method is accurate and competitive.

Fire spotting is often responsible for dangerous flare-ups in wildfires and causes secondary ignitions isolated from the primary fire zone, which lead to perilous situations. The main aim of the present research is to provide a versatile probabilistic model for fire spotting that is suitable for implementation as a post-processing scheme at each time step in any of the existing operational large-scale wildfire propagation models, without calling for any major changes in the original framework. In particular, a complete physical parameterisation of fire spotting is presented and the corresponding updated model RandomFront 2.3 is implemented in a coupled fire–atmosphere model: WRF-SFIRE. A test case is simulated and discussed. Moreover, the results from different simulations with a simple model based on the level set method, namely LSFire+, highlight the response of the parameterisation to varying fire intensities, wind conditions and different firebrand radii. The contribution of the firebrands to increasing the fire perimeter varies according to different concurrent conditions, and the simulations show results in agreement with the physical processes. Among the many rigorous approaches available in the literature to model firebrand transport and distribution, the approach presented here proves to be simple yet versatile for application to operational large-scale fire spread models.