Predictive Modeling of Urban Flooding Using Finite Differences and Numerical Integration

Urban flooding is a complex phenomenon driven by limitations regarding precipitation and drainage systems. In this sense, a dynamic analysis of the processes allows us to model them by describing water accumulation over time through integrals and analyzing the instantaneous rate of change through derivatives. This approach enables early identification of risk situations before critical levels are reached. Concepts of differential and integral calculus are applied to model water accumulation in urban environments and predict flood risks. The following work proposes using continuous and discrete mathematical modeling to monitor hydrological behavior. The theoretical foundation is based on three pillars: the definition of water accumulation A(t) as the integral of the difference between rainfall intensity R(t) and drainage capacity D(t). Computationally, the discrete model is A(t+delta t)=A(t)+ (R-D); the instantaneous rate of change A'(t)=R(t)-D(t) as the main risk indicator (positive values indicate increasing accumulation, while rates that exceed a critical threshold trigger accelerated risk alerts); and the application of limiting casesto represent extreme behaviors, such as rainfall intensity approaching or tdrainage capacity approaching its physical maximum. Simulations demonstrated that the application identifies critical points A'(t) = 0, and predicts flooding before the level reaches the safety threshold (Ac). Thus, the model demonstrates the effectiveness of applying mathematics for the analysis of environmental problems. Integrating functions, derivatives, and integrals into a modern platform enables the transformation of mathematical models into tools with high social impact.