A brief overview of modern research of the processes dynamics in unsteady water ows using the shallow water equation

Abstract

In hydrodynamics (hydraulics), there are numerous approaches to solving the problem of water flow dynamics control in river beds and channels, while the results of each methods differ, and estimates of their reliability do not always exist. The shallow water equation (or Saint-Venant's equations in one-dimensional form) is often used by hydraulic engineers in their practice. Its apparent simplicity and ability to describe well enough the behavior of rivers and flows make it a useful tool for many applications, such as the regulation of navigable rivers and irrigation networks in agriculture. The main direction of research in the field of numeric problems described by Saint-Venant equations is the development of numerical methods of computation implemented on super-powerful computers. Development of numerical models of surface water dynamics in the shallow water approximation is actively advancing during the recent years. The article is devoted to a review of mathematical studies of the dynamics of processes in unsteady water flows using differential equations, as well as an assessment of these approaches from the point of view of the model's reflection of real processes. The work is aimed at analyzing different approaches to modeling the dynamics of processes in non-stationary water flows. The objectives of the study include the analysis of scientific publications with different approaches to modeling the shallow water equation, taking into account factors, parameters, and modeling methods.

More by Zhanars Abdiramanov

• Fast Spectral Simulation of One-Dimensional Nonlinear Fractional Waves in Attenuating MediaComputation · 2026
• A Spectral–Physics-Informed Neural Network Hybrid with Energy-Constrained Training for the Allen–Cahn Equation: Forward and Inverse ProblemsPreprints.org · 2026
• Code and Lean 4 proofs for: Fast Spectral Simulation of One-Dimensional Nonlinear Fractional Waves in Attenuating MediaZenodo · 2026
• A Fast Chebyshev Spectral Collocation Method for a Coupled System of Nonlinear Klein–Gordon Equations with Caputo Fractional MemoryAxioms · 2026 · Citations: 1