Two-Dimensional Shallow Water Equations: A Comparative Study of Modern Approaches

The two-dimensional (2D) shallow water equations (SWEs) form the cornerstone of hydrodynamic modeling across coastal engineering, flood prediction, and atmospheric science. Over the past few decades, their numerical solution has evolved significantly, transitioning from first-order finite difference methods to advanced high-resolution, adaptive schemes. This study presents a comprehensive survey of classical and modern solvers developed for SWE, with a briefing of the post-2020 advances. Solvers are categorized into finite volume, discontinuous Galerkin, and finite element methods, highlighting their stability, accuracy, and treatment of complex topography and boundary conditions in related works. Moreover, this study explores the emerging role of wavelet-based adaptive algorithms, which offer multiscale resolution and computational efficiency, especially in resolving test cases of dam breaking, circular wave propagation, and tsunami-like Gaussian disturbance. Identification of recent hybrid approaches that integrate the benefits of classical schemes with wavelet basis is the key idea. By synthesizing developments, this article aims to guide researchers and practitioners in selecting and designing appropriate numerical strategies for 2D SWE. It outlines current challenges and directions for future research in adaptive, scalable, and real-time SWE simulation by indicating the accuracy of the approaches utilized.