Implicit Finite-Difference Scheme for Two-Dimensional Flood Modelling Using Shallow-Water Equations

Accurate and computationally efficient numerical modelling of shallow-water flows is essential for flood prediction and hydrodynamic risk assessment. This study develops an implicit finite-difference scheme for the numerical solution of the two-dimensional shallow-water equations. First, the main numerical approaches used for shallow-water modelling, including finite-difference, finite-volume, finite-element, and discontinuous Galerkin methods, are analysed in terms of accuracy, stability, treatment of discontinuities, and computational requirements. Based on this analysis, an implicit finite-difference formulation is developed that uses central approximations for spatial derivatives and averages flow variables at cell boundaries. The nonlinear terms are treated using Newton linearization, resulting in an iterative scheme that allows larger time steps than explicit formulations constrained by the Courant–Friedrichs–Lewy condition. The proposed method is implemented in MATLAB as a computational module for two-dimensional hydrodynamic simulations. Its performance is demonstrated on a test problem that describes the propagation of an initially localised disturbance in a rectangular computational domain with rigid boundaries. The numerical results demonstrate stable wave propagation, conservation of the modelled flow dynamics, and physically consistent boundary reflections. The developed approach provides a computational basis for further integration of shallow-water hydrodynamic models with spatial data and geographic information systems for flood forecasting and risk assessment. The implicit scheme allowed the release of time steps Δt = 0.1, 0.5 and 0.9, which significantly exceeds the limit stability of the explicit scheme, which, due to the Courant–Friedrichs–Lévy conditions, was limited to the value Δt ≤ 0.01. The simulation results show that the developed scheme provides stable wave growth and physically correct separation from impermeable boundaries for all investigated time step indicators. The obtained water depth profiles at times t = 10, 15, 20 and 25 s illustrate the correct evolution of the initial combustion: the wave front expands symmetrically while preserving the conservative properties of the model hydrodynamics. The numerical solution demonstrates the accuracy and robustness of the proposed implicit finite-difference scheme, even when using time steps that are almost two orders of magnitude larger than those allowed by explicit methods. The results confirm that the developed approach is a robust and computationally efficient tool for hydrodynamic modelling, intended for further integration with geographic information systems in behaviour prediction and risk assessment tasks.